Linear sandwich semigroups Igor Dolinka sc:ala seminar , 10 December 2015
The bad news. . . . . . all groups will be monoids with no identity. Igor Dolinka 1 Linear sandwich semigroups
The good news. . . . . . all sandwiches are vegan, halal, kosher, nut free, gluten free. . . Igor Dolinka 2 Linear sandwich semigroups
Joint work of: yours truly. . . Igor Dolinka 3 Linear sandwich semigroups
. . . and James East (FBI) Igor Dolinka 4 Linear sandwich semigroups
. . . errr, I mean, James East (Western Sydney University) Igor Dolinka 5 Linear sandwich semigroups
Sandwiches? Igor Dolinka 6 Linear sandwich semigroups
Sandwiches? Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955) Igor Dolinka 6 Linear sandwich semigroups
Sandwiches? Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955) ◮ Let M mn be the set of all m × n matrices over a field F. Igor Dolinka 6 Linear sandwich semigroups
Sandwiches? Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955) ◮ Let M mn be the set of all m × n matrices over a field F. ◮ Fix A ∈ M nm . Igor Dolinka 6 Linear sandwich semigroups
Sandwiches? Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955) ◮ Let M mn be the set of all m × n matrices over a field F. ◮ Fix A ∈ M nm . ◮ For X , Y ∈ M mn , define X ⋆ Y = XAY . Igor Dolinka 6 Linear sandwich semigroups
Sandwiches? Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955) ◮ Let M mn be the set of all m × n matrices over a field F. ◮ Fix A ∈ M nm . ◮ For X , Y ∈ M mn , define X ⋆ Y = XAY . ◮ M A mn = ( M mn , ⋆ ) is a linear sandwich semigroup. Igor Dolinka 6 Linear sandwich semigroups
Sandwiches? Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955) ◮ Let M mn be the set of all m × n matrices over a field F. ◮ Fix A ∈ M nm . ◮ For X , Y ∈ M mn , define X ⋆ Y = XAY . ◮ M A mn = ( M mn , ⋆ ) is a linear sandwich semigroup. Example If m = n and A = I , then M A mn = M n is the full linear monoid. Igor Dolinka 6 Linear sandwich semigroups
Plan (non-linear) Igor Dolinka 7 Linear sandwich semigroups
Plan (non-linear) ◮ Green’s relations Igor Dolinka 7 Linear sandwich semigroups
Plan (non-linear) ◮ Green’s relations ◮ regular elements Igor Dolinka 7 Linear sandwich semigroups
Plan (non-linear) ◮ Green’s relations ◮ regular elements ◮ ideals Igor Dolinka 7 Linear sandwich semigroups
Plan (non-linear) ◮ Green’s relations ◮ regular elements ◮ ideals ◮ idempotent generation Igor Dolinka 7 Linear sandwich semigroups
Plan (non-linear) ◮ Green’s relations ◮ regular elements ◮ ideals ◮ idempotent generation ◮ small (idempotent) generating sets Igor Dolinka 7 Linear sandwich semigroups
Plan (non-linear) ◮ Green’s relations ◮ regular elements ◮ ideals ◮ idempotent generation ◮ small (idempotent) generating sets ◮ bigger sandwiches? Igor Dolinka 7 Linear sandwich semigroups
Egg-box diagrams for M n ( F = Z 2 ) 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 Igor Dolinka 8 Linear sandwich semigroups
Egg-box diagrams for M n ( F = Z 2 ) 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 box rank Within a , matrices have same . row column space column row space Igor Dolinka 8 Linear sandwich semigroups
Egg-box diagrams for M n ( F = Z 2 ) 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 box D -related Within a , matrices are . row R -related column L -related Igor Dolinka 8 Linear sandwich semigroups
Egg-box diagrams for M n ( F = Z 2 ) 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 ◮ Cells with an idempotent matrix are shaded. ◮ These are subgroups of M n ( F ) isomorphic to GL( r , F ). Igor Dolinka 8 Linear sandwich semigroups
Egg sandwiches — M A mn 1 1 1 1 1 1 1 1 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 Igor Dolinka 9 Linear sandwich semigroups
Egg sandwiches — M A mn 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 Igor Dolinka 10 Linear sandwich semigroups
Easy lemmas Lemma mn ∼ If A , B ∈ M nm and rank( A ) = rank( B ), then M A = M B mn . Igor Dolinka 11 Linear sandwich semigroups
Easy lemmas Lemma mn ∼ If A , B ∈ M nm and rank( A ) = rank( B ), then M A = M B mn . ◮ So we study M A mn , where � I r O � J = J r = ∈ M nm (0 ≤ r ≤ min( m , n ) is fixed) . O O Igor Dolinka 11 Linear sandwich semigroups
Easy lemmas Lemma mn ∼ If A , B ∈ M nm and rank( A ) = rank( B ), then M A = M B mn . ◮ So we study M A mn , where � I r O � J = J r = ∈ M nm (0 ≤ r ≤ min( m , n ) is fixed) . O O ◮ Write elements of M mn in 2 × 2 block form: � A B � , where A ∈ M rr , B ∈ M r , n − r , etc. C D Igor Dolinka 11 Linear sandwich semigroups
Easy lemmas Lemma mn ∼ If A , B ∈ M nm and rank( A ) = rank( B ), then M A = M B mn . ◮ So we study M A mn , where � I r O � J = J r = ∈ M nm (0 ≤ r ≤ min( m , n ) is fixed) . O O ◮ Write elements of M mn in 2 × 2 block form: � A B � , where A ∈ M rr , B ∈ M r , n − r , etc. C D Lemma � A B � E F � AE AF � � � ⋆ = . C D G H CE CF Igor Dolinka 11 Linear sandwich semigroups
Green’s relations Igor Dolinka 12 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ X R Y ⇔ X M n = Y M n , Igor Dolinka 12 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ X R Y ⇔ X M n = Y M n , ◮ X L Y ⇔ M m X = M m Y , Igor Dolinka 12 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ X R Y ⇔ X M n = Y M n , ◮ X L Y ⇔ M m X = M m Y , ◮ X J Y ⇔ M m X M n = M m Y M n , Igor Dolinka 12 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ H = R ∩ L , ◮ X R Y ⇔ X M n = Y M n , ◮ D = R ∨ L . ◮ X L Y ⇔ M m X = M m Y , ◮ X J Y ⇔ M m X M n = M m Y M n , Igor Dolinka 12 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ H = R ∩ L , ◮ X R Y ⇔ X M n = Y M n , ◮ D = R ∨ L . ◮ X L Y ⇔ M m X = M m Y , ◮ X J Y ⇔ M m X M n = M m Y M n , For X ∈ M mn , write R X = { Y ∈ M mn : X R Y } , etc. Igor Dolinka 12 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ H = R ∩ L , ◮ X R Y ⇔ X M n = Y M n , ◮ D = R ∨ L . ◮ X L Y ⇔ M m X = M m Y , ◮ X J Y ⇔ M m X M n = M m Y M n , For X ∈ M mn , write R X = { Y ∈ M mn : X R Y } , etc. Proposition ◮ R X = { Y ∈ M mn : Col( X ) = Col( Y ) } , ◮ L X = { Y ∈ M mn : Row( X ) = Row( Y ) } , ◮ J X = D X = { Y ∈ M mn : rank( X ) = rank( Y ) } Igor Dolinka 12 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ X R Y ⇔ X ⋆ M mn = Y ⋆ M mn , — i.e., XJ M mn = YJ M mn , Igor Dolinka 13 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ X R Y ⇔ X ⋆ M mn = Y ⋆ M mn , — i.e., XJ M mn = YJ M mn , ◮ X L Y ⇔ M mn ⋆ X = M mn ⋆ Y , Igor Dolinka 13 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ X R Y ⇔ X ⋆ M mn = Y ⋆ M mn , — i.e., XJ M mn = YJ M mn , ◮ X L Y ⇔ M mn ⋆ X = M mn ⋆ Y , ◮ X J Y ⇔ M mn ⋆ X ⋆ M mn = M mn ⋆ Y ⋆ M mn , Igor Dolinka 13 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ X R Y ⇔ X ⋆ M mn = Y ⋆ M mn , — i.e., XJ M mn = YJ M mn , ◮ X L Y ⇔ M mn ⋆ X = M mn ⋆ Y , ◮ X J Y ⇔ M mn ⋆ X ⋆ M mn = M mn ⋆ Y ⋆ M mn , ◮ H = R ∩ L , ◮ D = R ∨ L . Igor Dolinka 13 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ X R Y ⇔ X ⋆ M mn = Y ⋆ M mn , — i.e., XJ M mn = YJ M mn , ◮ X L Y ⇔ M mn ⋆ X = M mn ⋆ Y , ◮ X J Y ⇔ M mn ⋆ X ⋆ M mn = M mn ⋆ Y ⋆ M mn , ◮ H = R ∩ L , ◮ D = R ∨ L . These are the usual Green’s relations on M A mn . Igor Dolinka 13 Linear sandwich semigroups
Green’s relations Let X , Y ∈ M mn . Write ◮ X R Y ⇔ X ⋆ M mn = Y ⋆ M mn , — i.e., XJ M mn = YJ M mn , ◮ X L Y ⇔ M mn ⋆ X = M mn ⋆ Y , ◮ X J Y ⇔ M mn ⋆ X ⋆ M mn = M mn ⋆ Y ⋆ M mn , ◮ H = R ∩ L , ◮ D = R ∨ L . These are the usual Green’s relations on M A mn . For X ∈ M mn , write R J X = { Y ∈ M mn : X R Y } , etc. Igor Dolinka 13 Linear sandwich semigroups
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