Spectrum problems for structures arising from lattices and rings - PowerPoint PPT Presentation

Spectral spaces Spectrum A nonempty closed set F in a topological space X is problems for structures irreducible if F = A ∪ B implies that either F = A or arising from lattices and F = B , for all closed sets A and B . rings We say that X is sober if every irreducible closed set is { x } (the closure of { x } ) for a unique x ∈ X . Hochster’s Theorem for ◦ Set K ( X ) = def { U ⊆ X | U is open and compact } . commutative unital rings ◦ ◦ Stone duality In general, U , V ∈ K ( X ) ⇒ U ∪ V ∈ K ( X ). for bounded distributive ◦ ◦ lattices However, usually U , V ∈ K ( X ) �⇒ U ∩ V ∈ K ( X ). ℓ -spectra of ◦ We say that X is spectral if it is sober and K ( X ) is a basis Abelian ℓ -groups of the topology of X , closed under finite intersection. The real Taking the empty intersection then yields that X is spectrum of a commutative, compact (usually not Hausdorff). unital ring Spectral Spec A is a spectral space, for every commutative unital scrummage ring A (well known and easy).

Hochster’s Theorem Spectrum The converse of the above observation holds: problems for structures arising from Theorem (Hochster 1969) lattices and rings Every spectral space X is homeomorphic to Spec A for some commutative unital ring A . Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Hochster’s Theorem Spectrum The converse of the above observation holds: problems for structures arising from Theorem (Hochster 1969) lattices and rings Every spectral space X is homeomorphic to Spec A for some commutative unital ring A . Hochster’s Theorem for commutative unital rings Moreover, Hochster proves that the assignment X �→ A Stone duality can be made functorial. for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Hochster’s Theorem Spectrum The converse of the above observation holds: problems for structures arising from Theorem (Hochster 1969) lattices and rings Every spectral space X is homeomorphic to Spec A for some commutative unital ring A . Hochster’s Theorem for commutative unital rings Moreover, Hochster proves that the assignment X �→ A Stone duality can be made functorial. for bounded distributive In order for that observation to make sense, the lattices morphisms need to be specified. ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Hochster’s Theorem Spectrum The converse of the above observation holds: problems for structures arising from Theorem (Hochster 1969) lattices and rings Every spectral space X is homeomorphic to Spec A for some commutative unital ring A . Hochster’s Theorem for commutative unital rings Moreover, Hochster proves that the assignment X �→ A Stone duality can be made functorial. for bounded distributive In order for that observation to make sense, the lattices morphisms need to be specified. ℓ -spectra of Abelian ℓ -groups On the ring side, just consider unital ring homomorphisms. The real spectrum of a commutative, unital ring Spectral scrummage

Hochster’s Theorem Spectrum The converse of the above observation holds: problems for structures arising from Theorem (Hochster 1969) lattices and rings Every spectral space X is homeomorphic to Spec A for some commutative unital ring A . Hochster’s Theorem for commutative unital rings Moreover, Hochster proves that the assignment X �→ A Stone duality can be made functorial. for bounded distributive In order for that observation to make sense, the lattices morphisms need to be specified. ℓ -spectra of Abelian ℓ -groups On the ring side, just consider unital ring homomorphisms. The real On the spectral space side, consider surjective spectral spectrum of a commutative, maps. unital ring Spectral scrummage

Hochster’s Theorem Spectrum The converse of the above observation holds: problems for structures arising from Theorem (Hochster 1969) lattices and rings Every spectral space X is homeomorphic to Spec A for some commutative unital ring A . Hochster’s Theorem for commutative unital rings Moreover, Hochster proves that the assignment X �→ A Stone duality can be made functorial. for bounded distributive In order for that observation to make sense, the lattices morphisms need to be specified. ℓ -spectra of Abelian ℓ -groups On the ring side, just consider unital ring homomorphisms. The real On the spectral space side, consider surjective spectral spectrum of a commutative, maps. For spectral spaces X and Y , a map f : X → Y is unital ring ◦ ◦ Spectral spectral if f − 1 [ V ] ∈ K ( X ) whenever V ∈ K ( Y ). scrummage

The spectrum of a bounded distributive lattice Spectrum problems for structures A subset I in a bounded distributive lattice D is an ideal arising from lattices and of D if 0 ∈ I , ( { x , y } ⊆ I ⇒ x ∨ y ∈ I ), and rings ( { x , y } ∩ I � = ∅ ⇒ x ∧ y ∈ I ). An ideal I is prime if I � = D and ( x ∧ y ∈ I ⇒ { x , y } ∩ I � = ∅ ). Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

The spectrum of a bounded distributive lattice Spectrum problems for structures A subset I in a bounded distributive lattice D is an ideal arising from lattices and of D if 0 ∈ I , ( { x , y } ⊆ I ⇒ x ∨ y ∈ I ), and rings ( { x , y } ∩ I � = ∅ ⇒ x ∧ y ∈ I ). An ideal I is prime if I � = D and ( x ∧ y ∈ I ⇒ { x , y } ∩ I � = ∅ ). Hochster’s Theorem for For a bounded distributive lattice D , set commutative unital rings Spec D = def { P | P is a prime ideal of D } , endowed with Stone duality the topology whose closed sets are the sets of the form for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

The spectrum of a bounded distributive lattice Spectrum problems for structures A subset I in a bounded distributive lattice D is an ideal arising from lattices and of D if 0 ∈ I , ( { x , y } ⊆ I ⇒ x ∨ y ∈ I ), and rings ( { x , y } ∩ I � = ∅ ⇒ x ∧ y ∈ I ). An ideal I is prime if I � = D and ( x ∧ y ∈ I ⇒ { x , y } ∩ I � = ∅ ). Hochster’s Theorem for For a bounded distributive lattice D , set commutative unital rings Spec D = def { P | P is a prime ideal of D } , endowed with Stone duality the topology whose closed sets are the sets of the form for bounded distributive lattices Spec( D , X ) = def { P ∈ Spec D | X ⊆ P } , for X ⊆ D , ℓ -spectra of Abelian ℓ -groups and we call it the spectrum of D . The real spectrum of a commutative, unital ring Spectral scrummage

The spectrum of a bounded distributive lattice Spectrum problems for structures A subset I in a bounded distributive lattice D is an ideal arising from lattices and of D if 0 ∈ I , ( { x , y } ⊆ I ⇒ x ∨ y ∈ I ), and rings ( { x , y } ∩ I � = ∅ ⇒ x ∧ y ∈ I ). An ideal I is prime if I � = D and ( x ∧ y ∈ I ⇒ { x , y } ∩ I � = ∅ ). Hochster’s Theorem for For a bounded distributive lattice D , set commutative unital rings Spec D = def { P | P is a prime ideal of D } , endowed with Stone duality the topology whose closed sets are the sets of the form for bounded distributive lattices Spec( D , X ) = def { P ∈ Spec D | X ⊆ P } , for X ⊆ D , ℓ -spectra of Abelian ℓ -groups and we call it the spectrum of D . The real spectrum of a commutative, It is well known that the spectrum of any bounded unital ring distributive lattice is a spectral space. Spectral scrummage

The functors underlying Stone duality Spectrum problems for For bounded distributive lattices D and E and a structures arising from 0 , 1-lattice homomorphism f : D → E , the map lattices and rings Spec f : Spec E → Spec D , Q �→ f − 1 [ Q ] is spectral. Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

The functors underlying Stone duality Spectrum problems for For bounded distributive lattices D and E and a structures arising from 0 , 1-lattice homomorphism f : D → E , the map lattices and rings Spec f : Spec E → Spec D , Q �→ f − 1 [ Q ] is spectral. For spectral spaces X and Y and a spectral map Hochster’s ◦ ◦ ◦ Theorem for K ( X ), V �→ ϕ − 1 [ V ] ϕ : X → Y , the map K ( ϕ ): K ( Y ) → commutative unital rings is a 0 , 1-lattice homomorphism. Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

The functors underlying Stone duality Spectrum problems for For bounded distributive lattices D and E and a structures arising from 0 , 1-lattice homomorphism f : D → E , the map lattices and rings Spec f : Spec E → Spec D , Q �→ f − 1 [ Q ] is spectral. For spectral spaces X and Y and a spectral map Hochster’s ◦ ◦ ◦ Theorem for K ( X ), V �→ ϕ − 1 [ V ] ϕ : X → Y , the map K ( ϕ ): K ( Y ) → commutative unital rings is a 0 , 1-lattice homomorphism. Stone duality for bounded Theorem (Stone 1938) distributive lattices ◦ The pair (Spec , K ) induces a (categorical) duality, between ℓ -spectra of Abelian bounded distributive lattices with 0 , 1-lattice homomorphisms ℓ -groups and spectral spaces with spectral maps. The real spectrum of a commutative, unital ring Spectral scrummage

The functors underlying Stone duality Spectrum problems for For bounded distributive lattices D and E and a structures arising from 0 , 1-lattice homomorphism f : D → E , the map lattices and rings Spec f : Spec E → Spec D , Q �→ f − 1 [ Q ] is spectral. For spectral spaces X and Y and a spectral map Hochster’s ◦ ◦ ◦ Theorem for K ( X ), V �→ ϕ − 1 [ V ] ϕ : X → Y , the map K ( ϕ ): K ( Y ) → commutative unital rings is a 0 , 1-lattice homomorphism. Stone duality for bounded Theorem (Stone 1938) distributive lattices ◦ The pair (Spec , K ) induces a (categorical) duality, between ℓ -spectra of Abelian bounded distributive lattices with 0 , 1-lattice homomorphisms ℓ -groups and spectral spaces with spectral maps. The real spectrum of a commutative, unital ring Note that in Hochster’s Theorem’s case, we do not obtain Spectral a duality (a ring is not determined by its spectrum). scrummage

Further spectra? Spectrum problems for structures arising from lattices and rings To summarize: spectral spaces are the same as spectra of commutative unital rings, and also spectra of bounded Hochster’s Theorem for distributive lattices. commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Further spectra? Spectrum problems for structures arising from lattices and rings To summarize: spectral spaces are the same as spectra of commutative unital rings, and also spectra of bounded Hochster’s Theorem for distributive lattices. commutative unital rings In the case of bounded distributive lattices, we obtain a Stone duality duality. for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Further spectra? Spectrum problems for structures arising from lattices and rings To summarize: spectral spaces are the same as spectra of commutative unital rings, and also spectra of bounded Hochster’s Theorem for distributive lattices. commutative unital rings In the case of bounded distributive lattices, we obtain a Stone duality duality. In the case of commutative unital rings, we do for bounded distributive not. lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Further spectra? Spectrum problems for structures arising from lattices and rings To summarize: spectral spaces are the same as spectra of commutative unital rings, and also spectra of bounded Hochster’s Theorem for distributive lattices. commutative unital rings In the case of bounded distributive lattices, we obtain a Stone duality duality. In the case of commutative unital rings, we do for bounded distributive not. lattices ℓ -spectra of Further algebraic structures also afford a concept of Abelian ℓ -groups spectrum. The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -ideals of an Abelian ℓ -group Spectrum An ℓ -group is a group endowed with a lattice ordering ≤ , problems for structures such that x ≤ y implies both xz ≤ yz and zx ≤ zy . arising from lattices and rings Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -ideals of an Abelian ℓ -group Spectrum An ℓ -group is a group endowed with a lattice ordering ≤ , problems for structures such that x ≤ y implies both xz ≤ yz and zx ≤ zy . arising from lattices and The underlying lattice of an ℓ -group is necessarily rings distributive. Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -ideals of an Abelian ℓ -group Spectrum An ℓ -group is a group endowed with a lattice ordering ≤ , problems for structures such that x ≤ y implies both xz ≤ yz and zx ≤ zy . arising from lattices and The underlying lattice of an ℓ -group is necessarily rings distributive. Our ℓ -groups will be Abelian ( xy = yx ), thus we will Hochster’s Theorem for denote them additively ( x + y = y + x , commutative unital rings G + = def { x ∈ G | x ≥ 0 } , | x | = def x ∨ ( − x )). Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -ideals of an Abelian ℓ -group Spectrum An ℓ -group is a group endowed with a lattice ordering ≤ , problems for structures such that x ≤ y implies both xz ≤ yz and zx ≤ zy . arising from lattices and The underlying lattice of an ℓ -group is necessarily rings distributive. Our ℓ -groups will be Abelian ( xy = yx ), thus we will Hochster’s Theorem for denote them additively ( x + y = y + x , commutative unital rings G + = def { x ∈ G | x ≥ 0 } , | x | = def x ∨ ( − x )). Stone duality for bounded An additive subgroup of an Abelian ℓ -group G is an ℓ -ideal distributive lattices if it is both order-convex and closed under x �→ | x | . ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -ideals of an Abelian ℓ -group Spectrum An ℓ -group is a group endowed with a lattice ordering ≤ , problems for structures such that x ≤ y implies both xz ≤ yz and zx ≤ zy . arising from lattices and The underlying lattice of an ℓ -group is necessarily rings distributive. Our ℓ -groups will be Abelian ( xy = yx ), thus we will Hochster’s Theorem for denote them additively ( x + y = y + x , commutative unital rings G + = def { x ∈ G | x ≥ 0 } , | x | = def x ∨ ( − x )). Stone duality for bounded An additive subgroup of an Abelian ℓ -group G is an ℓ -ideal distributive lattices if it is both order-convex and closed under x �→ | x | . ℓ -spectra of An ℓ -ideal I of G is Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -ideals of an Abelian ℓ -group Spectrum An ℓ -group is a group endowed with a lattice ordering ≤ , problems for structures such that x ≤ y implies both xz ≤ yz and zx ≤ zy . arising from lattices and The underlying lattice of an ℓ -group is necessarily rings distributive. Our ℓ -groups will be Abelian ( xy = yx ), thus we will Hochster’s Theorem for denote them additively ( x + y = y + x , commutative unital rings G + = def { x ∈ G | x ≥ 0 } , | x | = def x ∨ ( − x )). Stone duality for bounded An additive subgroup of an Abelian ℓ -group G is an ℓ -ideal distributive lattices if it is both order-convex and closed under x �→ | x | . ℓ -spectra of An ℓ -ideal I of G is Abelian ℓ -groups prime if I � = G and x ∧ y ∈ I ⇒ { x , y } ∩ I � = ∅ . The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -ideals of an Abelian ℓ -group Spectrum An ℓ -group is a group endowed with a lattice ordering ≤ , problems for structures such that x ≤ y implies both xz ≤ yz and zx ≤ zy . arising from lattices and The underlying lattice of an ℓ -group is necessarily rings distributive. Our ℓ -groups will be Abelian ( xy = yx ), thus we will Hochster’s Theorem for denote them additively ( x + y = y + x , commutative unital rings G + = def { x ∈ G | x ≥ 0 } , | x | = def x ∨ ( − x )). Stone duality for bounded An additive subgroup of an Abelian ℓ -group G is an ℓ -ideal distributive lattices if it is both order-convex and closed under x �→ | x | . ℓ -spectra of An ℓ -ideal I of G is Abelian ℓ -groups prime if I � = G and x ∧ y ∈ I ⇒ { x , y } ∩ I � = ∅ . The real finitely generated (equivalently, principal) if spectrum of a commutative, I = � a � = { x ∈ G | ( ∃ n )( | x | ≤ na ) } for some a ∈ G + . unital ring Spectral scrummage

ℓ -ideals of an Abelian ℓ -group Spectrum An ℓ -group is a group endowed with a lattice ordering ≤ , problems for structures such that x ≤ y implies both xz ≤ yz and zx ≤ zy . arising from lattices and The underlying lattice of an ℓ -group is necessarily rings distributive. Our ℓ -groups will be Abelian ( xy = yx ), thus we will Hochster’s Theorem for denote them additively ( x + y = y + x , commutative unital rings G + = def { x ∈ G | x ≥ 0 } , | x | = def x ∨ ( − x )). Stone duality for bounded An additive subgroup of an Abelian ℓ -group G is an ℓ -ideal distributive lattices if it is both order-convex and closed under x �→ | x | . ℓ -spectra of An ℓ -ideal I of G is Abelian ℓ -groups prime if I � = G and x ∧ y ∈ I ⇒ { x , y } ∩ I � = ∅ . The real finitely generated (equivalently, principal) if spectrum of a commutative, I = � a � = { x ∈ G | ( ∃ n )( | x | ≤ na ) } for some a ∈ G + . unital ring An order-unit of G is an element e ∈ G + such that Spectral scrummage G = � e � .

The ℓ -spectrum of an Abelian ℓ -group with unit Spectrum problems for structures arising from lattices and For an Abelian ℓ -group G with (order-)unit, we set rings Spec ℓ G = def { P | P is a prime ideal of G } , endowed with Hochster’s the topology whose closed sets are the sets of the form Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

The ℓ -spectrum of an Abelian ℓ -group with unit Spectrum problems for structures arising from lattices and For an Abelian ℓ -group G with (order-)unit, we set rings Spec ℓ G = def { P | P is a prime ideal of G } , endowed with Hochster’s the topology whose closed sets are the sets of the form Theorem for commutative unital rings Spec ℓ ( G , X ) = def { P ∈ Spec ℓ G | X ⊆ P } , for X ⊆ G , Stone duality for bounded distributive and we call it the ℓ -spectrum of G . lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

The ℓ -spectrum of an Abelian ℓ -group with unit Spectrum problems for structures arising from lattices and For an Abelian ℓ -group G with (order-)unit, we set rings Spec ℓ G = def { P | P is a prime ideal of G } , endowed with Hochster’s the topology whose closed sets are the sets of the form Theorem for commutative unital rings Spec ℓ ( G , X ) = def { P ∈ Spec ℓ G | X ⊆ P } , for X ⊆ G , Stone duality for bounded distributive and we call it the ℓ -spectrum of G . lattices ℓ -spectra of It is well known that the ℓ -spectrum of any Abelian Abelian ℓ -groups ℓ -group with unit is a spectral space. The real spectrum of a commutative, unital ring Spectral scrummage

The ℓ -spectrum of an Abelian ℓ -group with unit Spectrum problems for structures arising from lattices and For an Abelian ℓ -group G with (order-)unit, we set rings Spec ℓ G = def { P | P is a prime ideal of G } , endowed with Hochster’s the topology whose closed sets are the sets of the form Theorem for commutative unital rings Spec ℓ ( G , X ) = def { P ∈ Spec ℓ G | X ⊆ P } , for X ⊆ G , Stone duality for bounded distributive and we call it the ℓ -spectrum of G . lattices ℓ -spectra of It is well known that the ℓ -spectrum of any Abelian Abelian ℓ -groups ℓ -group with unit is a spectral space. The real It turns out that more is true! spectrum of a commutative, unital ring Spectral scrummage

Completely normal spectral spaces Spectrum In any topological space X , the specialization preordering problems for structures is defined by x � y if y ∈ { x } . arising from lattices and rings Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Completely normal spectral spaces Spectrum In any topological space X , the specialization preordering problems for structures is defined by x � y if y ∈ { x } . arising from lattices and If X is spectral (or, much more generally, if X is T 0 ), rings then � is an ordering (i.e., x � y and y � x implies that x = y ). Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Completely normal spectral spaces Spectrum In any topological space X , the specialization preordering problems for structures is defined by x � y if y ∈ { x } . arising from lattices and If X is spectral (or, much more generally, if X is T 0 ), rings then � is an ordering (i.e., x � y and y � x implies that x = y ). Hochster’s Theorem for A spectral space X is completely normal if � is a root commutative unital rings system, that is, { x , y } ⊆ { z } ⇒ ( x ∈ { y } or y ∈ { x } ). Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Completely normal spectral spaces Spectrum In any topological space X , the specialization preordering problems for structures is defined by x � y if y ∈ { x } . arising from lattices and If X is spectral (or, much more generally, if X is T 0 ), rings then � is an ordering (i.e., x � y and y � x implies that x = y ). Hochster’s Theorem for A spectral space X is completely normal if � is a root commutative unital rings system, that is, { x , y } ⊆ { z } ⇒ ( x ∈ { y } or y ∈ { x } ). Stone duality This is (properly) weaker than saying that every subspace for bounded distributive of X is normal. lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Completely normal spectral spaces Spectrum In any topological space X , the specialization preordering problems for structures is defined by x � y if y ∈ { x } . arising from lattices and If X is spectral (or, much more generally, if X is T 0 ), rings then � is an ordering (i.e., x � y and y � x implies that x = y ). Hochster’s Theorem for A spectral space X is completely normal if � is a root commutative unital rings system, that is, { x , y } ⊆ { z } ⇒ ( x ∈ { y } or y ∈ { x } ). Stone duality This is (properly) weaker than saying that every subspace for bounded distributive of X is normal. lattices ℓ -spectra of Theorem (Monteiro 1954) Abelian ℓ -groups A spectral space X is completely normal iff its Stone The real ◦ spectrum of a dual K ( X ) is a completely normal lattice, that is, commutative, unital ring Spectral scrummage

Completely normal spectral spaces Spectrum In any topological space X , the specialization preordering problems for structures is defined by x � y if y ∈ { x } . arising from lattices and If X is spectral (or, much more generally, if X is T 0 ), rings then � is an ordering (i.e., x � y and y � x implies that x = y ). Hochster’s Theorem for A spectral space X is completely normal if � is a root commutative unital rings system, that is, { x , y } ⊆ { z } ⇒ ( x ∈ { y } or y ∈ { x } ). Stone duality This is (properly) weaker than saying that every subspace for bounded distributive of X is normal. lattices ℓ -spectra of Theorem (Monteiro 1954) Abelian ℓ -groups A spectral space X is completely normal iff its Stone The real ◦ spectrum of a dual K ( X ) is a completely normal lattice, that is, commutative, unital ring Spectral ( ∀ a , b )( ∃ x , y )( a ∨ b = a ∨ y = x ∨ b and x ∧ y = 0) . scrummage

ℓ -spectra of Abelian ℓ -groups again Spectrum problems for Theorem (Keimel 1971) structures arising from The ℓ -spectrum of any Abelian ℓ -group with unit is a lattices and rings completely normal spectral space. Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -spectra of Abelian ℓ -groups again Spectrum problems for Theorem (Keimel 1971) structures arising from The ℓ -spectrum of any Abelian ℓ -group with unit is a lattices and rings completely normal spectral space. Hochster’s Theorem for The question, of characterizing ℓ -spectra, is open since commutative then. unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -spectra of Abelian ℓ -groups again Spectrum problems for Theorem (Keimel 1971) structures arising from The ℓ -spectrum of any Abelian ℓ -group with unit is a lattices and rings completely normal spectral space. Hochster’s Theorem for The question, of characterizing ℓ -spectra, is open since commutative then. unital rings Stone duality Equivalent to the MV-spectrum problem. for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -spectra of Abelian ℓ -groups again Spectrum problems for Theorem (Keimel 1971) structures arising from The ℓ -spectrum of any Abelian ℓ -group with unit is a lattices and rings completely normal spectral space. Hochster’s Theorem for The question, of characterizing ℓ -spectra, is open since commutative then. unital rings Stone duality Equivalent to the MV-spectrum problem. for bounded distributive lattices Theorem (Delzell and Madden 1994) ℓ -spectra of Abelian ℓ -groups Not every completely normal spectral space is an ℓ -spectrum. The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -spectra of Abelian ℓ -groups again Spectrum problems for Theorem (Keimel 1971) structures arising from The ℓ -spectrum of any Abelian ℓ -group with unit is a lattices and rings completely normal spectral space. Hochster’s Theorem for The question, of characterizing ℓ -spectra, is open since commutative then. unital rings Stone duality Equivalent to the MV-spectrum problem. for bounded distributive lattices Theorem (Delzell and Madden 1994) ℓ -spectra of Abelian ℓ -groups Not every completely normal spectral space is an ℓ -spectrum. The real spectrum of a Delzell and Madden’s example is not second countable (i.e., no commutative, unital ring countable basis of the topology): in fact, it has Spectral ◦ scrummage card K ( X ) = ℵ 1 .

ℓ -spectra of countable Abelian ℓ -groups Spectrum Theorem (W. 2017) problems for structures arising from Every second countable completely normal spectral space is lattices and rings homeomorphic to Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -spectra of countable Abelian ℓ -groups Spectrum Theorem (W. 2017) problems for structures arising from Every second countable completely normal spectral space is lattices and rings homeomorphic to Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s Theorem for Hence, Delzell and Madden’s counterexample cannot be commutative unital rings extended to the countable case. Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -spectra of countable Abelian ℓ -groups Spectrum Theorem (W. 2017) problems for structures arising from Every second countable completely normal spectral space is lattices and rings homeomorphic to Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s Theorem for Hence, Delzell and Madden’s counterexample cannot be commutative unital rings extended to the countable case. Stone duality Very rough outline of proof (of the countable case): start for bounded distributive by observing that for any Abelian ℓ -group G with unit, the lattices Stone dual of Spec ℓ G is Id c G , the lattice of all principal ℓ -spectra of Abelian ℓ -ideals of G (ordered by ⊆ ). ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

ℓ -spectra of countable Abelian ℓ -groups Spectrum Theorem (W. 2017) problems for structures arising from Every second countable completely normal spectral space is lattices and rings homeomorphic to Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s Theorem for Hence, Delzell and Madden’s counterexample cannot be commutative unital rings extended to the countable case. Stone duality Very rough outline of proof (of the countable case): start for bounded distributive by observing that for any Abelian ℓ -group G with unit, the lattices Stone dual of Spec ℓ G is Id c G , the lattice of all principal ℓ -spectra of Abelian ℓ -ideals of G (ordered by ⊆ ). ℓ -groups Since G has an order-unit, Id c G is a bounded distributive The real spectrum of a lattice. commutative, unital ring Spectral scrummage

ℓ -spectra of countable Abelian ℓ -groups Spectrum Theorem (W. 2017) problems for structures arising from Every second countable completely normal spectral space is lattices and rings homeomorphic to Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s Theorem for Hence, Delzell and Madden’s counterexample cannot be commutative unital rings extended to the countable case. Stone duality Very rough outline of proof (of the countable case): start for bounded distributive by observing that for any Abelian ℓ -group G with unit, the lattices Stone dual of Spec ℓ G is Id c G , the lattice of all principal ℓ -spectra of Abelian ℓ -ideals of G (ordered by ⊆ ). ℓ -groups Since G has an order-unit, Id c G is a bounded distributive The real spectrum of a lattice. commutative, unital ring Thus we must prove that every countable completely Spectral normal bounded distributive lattice D is ∼ = Id c G for some scrummage Abelian ℓ -group G with unit.

Very rough outline of the proof of the countable case (cont’d) Spectrum The idea is to construct a “nice” surjective 0 , 1-lattice problems for structures homomorphism f : Id c F ω ։ D , where F ω denotes the free arising from lattices and Abelian ℓ -group on a countably infinite generating set. rings Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (cont’d) Spectrum The idea is to construct a “nice” surjective 0 , 1-lattice problems for structures homomorphism f : Id c F ω ։ D , where F ω denotes the free arising from lattices and Abelian ℓ -group on a countably infinite generating set. rings “Nice” means that f should induce an isomorphism Hochster’s Id c ( F ω / I ) → D , for the ℓ -ideal I = def { x ∈ F ω | f ( � x � ) = 0 } . Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (cont’d) Spectrum The idea is to construct a “nice” surjective 0 , 1-lattice problems for structures homomorphism f : Id c F ω ։ D , where F ω denotes the free arising from lattices and Abelian ℓ -group on a countably infinite generating set. rings “Nice” means that f should induce an isomorphism Hochster’s Id c ( F ω / I ) → D , for the ℓ -ideal I = def { x ∈ F ω | f ( � x � ) = 0 } . Theorem for commutative unital rings It turns out that “nice” is easy to define! Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (cont’d) Spectrum The idea is to construct a “nice” surjective 0 , 1-lattice problems for structures homomorphism f : Id c F ω ։ D , where F ω denotes the free arising from lattices and Abelian ℓ -group on a countably infinite generating set. rings “Nice” means that f should induce an isomorphism Hochster’s Id c ( F ω / I ) → D , for the ℓ -ideal I = def { x ∈ F ω | f ( � x � ) = 0 } . Theorem for commutative unital rings It turns out that “nice” is easy to define! Stone duality for bounded distributive Definition (closed maps) lattices ℓ -spectra of For bounded distributive lattices A and B , a 0 , 1-lattice Abelian ℓ -groups homomorphism f : A → B is closed if whenever a 0 , a 1 ∈ A and The real b ∈ B , if f ( a 0 ) ≤ f ( a 1 ) ∨ b , then there exists x ∈ A such that spectrum of a commutative, a 0 ≤ a 1 ∨ x and f ( x ) ≤ b . unital ring Spectral scrummage

Very rough outline of the proof of the countable case (cont’d) Spectrum The idea is to construct a “nice” surjective 0 , 1-lattice problems for structures homomorphism f : Id c F ω ։ D , where F ω denotes the free arising from lattices and Abelian ℓ -group on a countably infinite generating set. rings “Nice” means that f should induce an isomorphism Hochster’s Id c ( F ω / I ) → D , for the ℓ -ideal I = def { x ∈ F ω | f ( � x � ) = 0 } . Theorem for commutative unital rings It turns out that “nice” is easy to define! Stone duality for bounded distributive Definition (closed maps) lattices ℓ -spectra of For bounded distributive lattices A and B , a 0 , 1-lattice Abelian ℓ -groups homomorphism f : A → B is closed if whenever a 0 , a 1 ∈ A and The real b ∈ B , if f ( a 0 ) ≤ f ( a 1 ) ∨ b , then there exists x ∈ A such that spectrum of a commutative, a 0 ≤ a 1 ∨ x and f ( x ) ≤ b . Equivalently, the Stone dual map unital ring Spec f : Spec B → Spec A is closed (i.e., it sends closed subsets Spectral scrummage to closed subsets).

Very rough outline of the proof of the countable case (further cont’d) Spectrum The map f : Id c F ω → D is constructed as f = � n <ω f n problems for structures (each f n ⊆ f n +1 ), where each f n : L n → D is a lattice arising from lattices and homomorphism, for a carefully constructed finite rings sublattice L n of Id c F ω . Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (further cont’d) Spectrum The map f : Id c F ω → D is constructed as f = � n <ω f n problems for structures (each f n ⊆ f n +1 ), where each f n : L n → D is a lattice arising from lattices and homomorphism, for a carefully constructed finite rings sublattice L n of Id c F ω . Due to a 2004 example of Di Nola and Grigolia, the L n Hochster’s Theorem for cannot all be completely normal. commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (further cont’d) Spectrum The map f : Id c F ω → D is constructed as f = � n <ω f n problems for structures (each f n ⊆ f n +1 ), where each f n : L n → D is a lattice arising from lattices and homomorphism, for a carefully constructed finite rings sublattice L n of Id c F ω . Due to a 2004 example of Di Nola and Grigolia, the L n Hochster’s Theorem for cannot all be completely normal. commutative unital rings The finite distributive lattices L n come out as special cases Stone duality of the following construction. for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (further cont’d) Spectrum The map f : Id c F ω → D is constructed as f = � n <ω f n problems for structures (each f n ⊆ f n +1 ), where each f n : L n → D is a lattice arising from lattices and homomorphism, for a carefully constructed finite rings sublattice L n of Id c F ω . Due to a 2004 example of Di Nola and Grigolia, the L n Hochster’s Theorem for cannot all be completely normal. commutative unital rings The finite distributive lattices L n come out as special cases Stone duality of the following construction. for bounded distributive Let H be a set of closed hyperplanes of a topological lattices vector space E . ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (further cont’d) Spectrum The map f : Id c F ω → D is constructed as f = � n <ω f n problems for structures (each f n ⊆ f n +1 ), where each f n : L n → D is a lattice arising from lattices and homomorphism, for a carefully constructed finite rings sublattice L n of Id c F ω . Due to a 2004 example of Di Nola and Grigolia, the L n Hochster’s Theorem for cannot all be completely normal. commutative unital rings The finite distributive lattices L n come out as special cases Stone duality of the following construction. for bounded distributive Let H be a set of closed hyperplanes of a topological lattices vector space E . ℓ -spectra of Abelian Each H ∈ H determines two open half-spaces H + and H − . ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (further cont’d) Spectrum The map f : Id c F ω → D is constructed as f = � n <ω f n problems for structures (each f n ⊆ f n +1 ), where each f n : L n → D is a lattice arising from lattices and homomorphism, for a carefully constructed finite rings sublattice L n of Id c F ω . Due to a 2004 example of Di Nola and Grigolia, the L n Hochster’s Theorem for cannot all be completely normal. commutative unital rings The finite distributive lattices L n come out as special cases Stone duality of the following construction. for bounded distributive Let H be a set of closed hyperplanes of a topological lattices vector space E . ℓ -spectra of Abelian Each H ∈ H determines two open half-spaces H + and H − . ℓ -groups Denote by Op( H ) the 0 , 1-sublattice of the powerset of E The real spectrum of a generated by { H + | H ∈ H } ∪ { H − | H ∈ H } . commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (further cont’d) Spectrum The map f : Id c F ω → D is constructed as f = � n <ω f n problems for structures (each f n ⊆ f n +1 ), where each f n : L n → D is a lattice arising from lattices and homomorphism, for a carefully constructed finite rings sublattice L n of Id c F ω . Due to a 2004 example of Di Nola and Grigolia, the L n Hochster’s Theorem for cannot all be completely normal. commutative unital rings The finite distributive lattices L n come out as special cases Stone duality of the following construction. for bounded distributive Let H be a set of closed hyperplanes of a topological lattices vector space E . ℓ -spectra of Abelian Each H ∈ H determines two open half-spaces H + and H − . ℓ -groups Denote by Op( H ) the 0 , 1-sublattice of the powerset of E The real spectrum of a generated by { H + | H ∈ H } ∪ { H − | H ∈ H } . commutative, unital ring def Op( H ) \ { E } is a sublattice The subset Op − ( H ) = Spectral scrummage of Op( H ).

Very rough outline of the proof of the countable case (coming to the end) Spectrum The lattices L n will have the form Op − ( H ), for finite sets problems for structures def R ( ω ) . of integer hyperplanes in E = arising from lattices and rings Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (coming to the end) Spectrum The lattices L n will have the form Op − ( H ), for finite sets problems for structures def R ( ω ) . of integer hyperplanes in E = arising from lattices and rings This is made possible by the Baker-Beynon duality, which implies that Id c F ω ∼ = Op − ( H Z ), where H Z denotes the Hochster’s (countable) set of all integer hyperplanes of R ( ω ) . Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (coming to the end) Spectrum The lattices L n will have the form Op − ( H ), for finite sets problems for structures def R ( ω ) . of integer hyperplanes in E = arising from lattices and rings This is made possible by the Baker-Beynon duality, which implies that Id c F ω ∼ = Op − ( H Z ), where H Z denotes the Hochster’s (countable) set of all integer hyperplanes of R ( ω ) . Theorem for commutative Each enlargement step, from f n to f n +1 , corrects one of unital rings the following three types of defects: Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (coming to the end) Spectrum The lattices L n will have the form Op − ( H ), for finite sets problems for structures def R ( ω ) . of integer hyperplanes in E = arising from lattices and rings This is made possible by the Baker-Beynon duality, which implies that Id c F ω ∼ = Op − ( H Z ), where H Z denotes the Hochster’s (countable) set of all integer hyperplanes of R ( ω ) . Theorem for commutative Each enlargement step, from f n to f n +1 , corrects one of unital rings the following three types of defects: Stone duality for bounded (hard) f n is not defined everywhere: then add a pair distributive lattices ( H + , H − ) to the domain of f n ; ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (coming to the end) Spectrum The lattices L n will have the form Op − ( H ), for finite sets problems for structures def R ( ω ) . of integer hyperplanes in E = arising from lattices and rings This is made possible by the Baker-Beynon duality, which implies that Id c F ω ∼ = Op − ( H Z ), where H Z denotes the Hochster’s (countable) set of all integer hyperplanes of R ( ω ) . Theorem for commutative Each enlargement step, from f n to f n +1 , corrects one of unital rings the following three types of defects: Stone duality for bounded (hard) f n is not defined everywhere: then add a pair distributive lattices ( H + , H − ) to the domain of f n ; ℓ -spectra of (easy, but infinite dimension needed!) f n is not surjective: Abelian ℓ -groups then add an element to the range of f n ; The real spectrum of a commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (coming to the end) Spectrum The lattices L n will have the form Op − ( H ), for finite sets problems for structures def R ( ω ) . of integer hyperplanes in E = arising from lattices and rings This is made possible by the Baker-Beynon duality, which implies that Id c F ω ∼ = Op − ( H Z ), where H Z denotes the Hochster’s (countable) set of all integer hyperplanes of R ( ω ) . Theorem for commutative Each enlargement step, from f n to f n +1 , corrects one of unital rings the following three types of defects: Stone duality for bounded (hard) f n is not defined everywhere: then add a pair distributive lattices ( H + , H − ) to the domain of f n ; ℓ -spectra of (easy, but infinite dimension needed!) f n is not surjective: Abelian ℓ -groups then add an element to the range of f n ; The real (hardest) f n is not closed: then let f n +1 correct a closure spectrum of a defect f n ( A 0 ) ≤ f n ( A 1 ) ∨ γ . commutative, unital ring Spectral scrummage

Very rough outline of the proof of the countable case (coming to the end) Spectrum The lattices L n will have the form Op − ( H ), for finite sets problems for structures def R ( ω ) . of integer hyperplanes in E = arising from lattices and rings This is made possible by the Baker-Beynon duality, which implies that Id c F ω ∼ = Op − ( H Z ), where H Z denotes the Hochster’s (countable) set of all integer hyperplanes of R ( ω ) . Theorem for commutative Each enlargement step, from f n to f n +1 , corrects one of unital rings the following three types of defects: Stone duality for bounded (hard) f n is not defined everywhere: then add a pair distributive lattices ( H + , H − ) to the domain of f n ; ℓ -spectra of (easy, but infinite dimension needed!) f n is not surjective: Abelian ℓ -groups then add an element to the range of f n ; The real (hardest) f n is not closed: then let f n +1 correct a closure spectrum of a defect f n ( A 0 ) ≤ f n ( A 1 ) ∨ γ . commutative, unital ring A crucial observation is that each Op( H ) is a Heyting Spectral scrummage subalgebra of the Heyting algebra of all open subsets of E .

Loose ends on ℓ -spectra Spectrum Say that a lattice D is ℓ -representable if it is ∼ = Id c G for problems for structures some Abelian ℓ -group G with unit. arising from lattices and rings Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Loose ends on ℓ -spectra Spectrum Say that a lattice D is ℓ -representable if it is ∼ = Id c G for problems for structures some Abelian ℓ -group G with unit. arising from lattices and rings Equivalently, D is the Stone dual of Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Loose ends on ℓ -spectra Spectrum Say that a lattice D is ℓ -representable if it is ∼ = Id c G for problems for structures some Abelian ℓ -group G with unit. arising from lattices and rings Equivalently, D is the Stone dual of Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s By the above, a countable bounded distributive lattice is Theorem for commutative ℓ -representable iff it is completely normal. unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Loose ends on ℓ -spectra Spectrum Say that a lattice D is ℓ -representable if it is ∼ = Id c G for problems for structures some Abelian ℓ -group G with unit. arising from lattices and rings Equivalently, D is the Stone dual of Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s By the above, a countable bounded distributive lattice is Theorem for commutative ℓ -representable iff it is completely normal. unital rings By Delzell and Madden’s example, this fails for Stone duality for bounded uncountable lattices. In fact, distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Loose ends on ℓ -spectra Spectrum Say that a lattice D is ℓ -representable if it is ∼ = Id c G for problems for structures some Abelian ℓ -group G with unit. arising from lattices and rings Equivalently, D is the Stone dual of Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s By the above, a countable bounded distributive lattice is Theorem for commutative ℓ -representable iff it is completely normal. unital rings By Delzell and Madden’s example, this fails for Stone duality for bounded uncountable lattices. In fact, distributive lattices ℓ -spectra of Theorem (W. 2017) Abelian ℓ -groups The class of all ℓ -representable lattices is not L ∞ ,ω -definable The real spectrum of a commutative, unital ring Spectral scrummage

Loose ends on ℓ -spectra Spectrum Say that a lattice D is ℓ -representable if it is ∼ = Id c G for problems for structures some Abelian ℓ -group G with unit. arising from lattices and rings Equivalently, D is the Stone dual of Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s By the above, a countable bounded distributive lattice is Theorem for commutative ℓ -representable iff it is completely normal. unital rings By Delzell and Madden’s example, this fails for Stone duality for bounded uncountable lattices. In fact, distributive lattices ℓ -spectra of Theorem (W. 2017) Abelian ℓ -groups The class of all ℓ -representable lattices is not L ∞ ,ω -definable The real (thus, a fortiori , not first-order definable). spectrum of a commutative, unital ring Spectral scrummage

Loose ends on ℓ -spectra Spectrum Say that a lattice D is ℓ -representable if it is ∼ = Id c G for problems for structures some Abelian ℓ -group G with unit. arising from lattices and rings Equivalently, D is the Stone dual of Spec ℓ G for some Abelian ℓ -group G with unit. Hochster’s By the above, a countable bounded distributive lattice is Theorem for commutative ℓ -representable iff it is completely normal. unital rings By Delzell and Madden’s example, this fails for Stone duality for bounded uncountable lattices. In fact, distributive lattices ℓ -spectra of Theorem (W. 2017) Abelian ℓ -groups The class of all ℓ -representable lattices is not L ∞ ,ω -definable The real (thus, a fortiori , not first-order definable). spectrum of a commutative, unital ring Analogous result for L ∞ ,λ (for any infinite cardinal λ ): Spectral scrummage proof currently under verification.

Cones, prime cones, real spectrum Spectrum The real spectrum was introduced in 1981, by Coste and problems for structures arising from Coste-Roy, as an ordered analogue of the Zariski spectrum lattices and of a commutative unital ring. rings Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Cones, prime cones, real spectrum Spectrum The real spectrum was introduced in 1981, by Coste and problems for structures arising from Coste-Roy, as an ordered analogue of the Zariski spectrum lattices and of a commutative unital ring. rings Let A be a commutative unital ring (not necessarily Hochster’s ordered). A cone of A is a subset C of A such that Theorem for C + C ⊆ C , C · C ⊆ C , and a 2 ∈ C whenever a ∈ A . commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Cones, prime cones, real spectrum Spectrum The real spectrum was introduced in 1981, by Coste and problems for structures arising from Coste-Roy, as an ordered analogue of the Zariski spectrum lattices and of a commutative unital ring. rings Let A be a commutative unital ring (not necessarily Hochster’s ordered). A cone of A is a subset C of A such that Theorem for C + C ⊆ C , C · C ⊆ C , and a 2 ∈ C whenever a ∈ A . commutative unital rings A cone C is prime if C ∩ ( − C ) is a prime ideal of A and Stone duality for bounded A = C ∪ ( − C ). distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Cones, prime cones, real spectrum Spectrum The real spectrum was introduced in 1981, by Coste and problems for structures arising from Coste-Roy, as an ordered analogue of the Zariski spectrum lattices and of a commutative unital ring. rings Let A be a commutative unital ring (not necessarily Hochster’s ordered). A cone of A is a subset C of A such that Theorem for C + C ⊆ C , C · C ⊆ C , and a 2 ∈ C whenever a ∈ A . commutative unital rings A cone C is prime if C ∩ ( − C ) is a prime ideal of A and Stone duality for bounded A = C ∪ ( − C ). distributive lattices We endow the set Spec r A of all prime cones of A with the ℓ -spectra of Abelian topology generated by the sets { P ∈ Spec r A | a / ∈ P } , for ℓ -groups a ∈ A . The real spectrum of a commutative, unital ring Spectral scrummage

Cones, prime cones, real spectrum Spectrum The real spectrum was introduced in 1981, by Coste and problems for structures arising from Coste-Roy, as an ordered analogue of the Zariski spectrum lattices and of a commutative unital ring. rings Let A be a commutative unital ring (not necessarily Hochster’s ordered). A cone of A is a subset C of A such that Theorem for C + C ⊆ C , C · C ⊆ C , and a 2 ∈ C whenever a ∈ A . commutative unital rings A cone C is prime if C ∩ ( − C ) is a prime ideal of A and Stone duality for bounded A = C ∪ ( − C ). distributive lattices We endow the set Spec r A of all prime cones of A with the ℓ -spectra of Abelian topology generated by the sets { P ∈ Spec r A | a / ∈ P } , for ℓ -groups a ∈ A . The topological space thus obtained is called the The real spectrum of a real spectrum of A . commutative, unital ring Spectral scrummage

Cones, prime cones, real spectrum Spectrum The real spectrum was introduced in 1981, by Coste and problems for structures arising from Coste-Roy, as an ordered analogue of the Zariski spectrum lattices and of a commutative unital ring. rings Let A be a commutative unital ring (not necessarily Hochster’s ordered). A cone of A is a subset C of A such that Theorem for C + C ⊆ C , C · C ⊆ C , and a 2 ∈ C whenever a ∈ A . commutative unital rings A cone C is prime if C ∩ ( − C ) is a prime ideal of A and Stone duality for bounded A = C ∪ ( − C ). distributive lattices We endow the set Spec r A of all prime cones of A with the ℓ -spectra of Abelian topology generated by the sets { P ∈ Spec r A | a / ∈ P } , for ℓ -groups a ∈ A . The topological space thus obtained is called the The real spectrum of a real spectrum of A . commutative, unital ring It turns out that Spec r A is a completely normal spectral Spectral space, for any commutative unital ring A . scrummage

Characterizing problem of real spectra Spectrum problems for Problem (Keimel 1991) structures arising from Characterize real spectra of commutative unital rings. lattices and rings Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Characterizing problem of real spectra Spectrum problems for Problem (Keimel 1991) structures arising from Characterize real spectra of commutative unital rings. lattices and rings The countable case of the problem above (i.e., for second Hochster’s Theorem for countable spaces) is still open. commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Characterizing problem of real spectra Spectrum problems for Problem (Keimel 1991) structures arising from Characterize real spectra of commutative unital rings. lattices and rings The countable case of the problem above (i.e., for second Hochster’s Theorem for countable spaces) is still open. commutative unital rings Negative answer in the uncountable case: Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Characterizing problem of real spectra Spectrum problems for Problem (Keimel 1991) structures arising from Characterize real spectra of commutative unital rings. lattices and rings The countable case of the problem above (i.e., for second Hochster’s Theorem for countable spaces) is still open. commutative unital rings Negative answer in the uncountable case: Stone duality for bounded distributive Theorem (Delzell and Madden 1994) lattices ℓ -spectra of Not every completely normal spectral space is a real spectrum. Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Characterizing problem of real spectra Spectrum problems for Problem (Keimel 1991) structures arising from Characterize real spectra of commutative unital rings. lattices and rings The countable case of the problem above (i.e., for second Hochster’s Theorem for countable spaces) is still open. commutative unital rings Negative answer in the uncountable case: Stone duality for bounded distributive Theorem (Delzell and Madden 1994) lattices ℓ -spectra of Not every completely normal spectral space is a real spectrum. Abelian ℓ -groups The real Theorem (Mellor and Tressl 2012) spectrum of a commutative, unital ring For any infinite cardinal λ , there is no L ∞ ,λ -characterization of Spectral the Stone duals of real spectra of commutative unital rings. scrummage

Subspaces of ℓ -spectra and real spectra Spectrum problems for structures arising from It is known that every closed subspace of an ℓ -spectrum (resp., lattices and rings real spectrum) is an ℓ -spectrum (resp., real spectrum). Hochster’s Theorem for commutative unital rings Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Subspaces of ℓ -spectra and real spectra Spectrum problems for structures arising from It is known that every closed subspace of an ℓ -spectrum (resp., lattices and rings real spectrum) is an ℓ -spectrum (resp., real spectrum). Hochster’s Theorem (W. 2017) Theorem for commutative Not every spectral subspace of an ℓ -spectrum (resp., real unital rings spectrum) is an ℓ -spectrum (resp., real spectrum). Stone duality for bounded distributive lattices ℓ -spectra of Abelian ℓ -groups The real spectrum of a commutative, unital ring Spectral scrummage

Subspaces of ℓ -spectra and real spectra Spectrum problems for structures arising from It is known that every closed subspace of an ℓ -spectrum (resp., lattices and rings real spectrum) is an ℓ -spectrum (resp., real spectrum). Hochster’s Theorem (W. 2017) Theorem for commutative Not every spectral subspace of an ℓ -spectrum (resp., real unital rings spectrum) is an ℓ -spectrum (resp., real spectrum). Stone duality for bounded distributive lattices Problem (W. 2017) ℓ -spectra of Abelian ℓ -groups Is a retract of an ℓ -spectrum also an ℓ -spectrum? Same The real question for real spectra. spectrum of a commutative, unital ring Spectral scrummage