Inspirations and objectives Half-plane structures Oval structures Two systems of point-free affine geometry Giangiacomo Gerla 1 nski 2 Rafał Gruszczy´ 1 IIASS University of Salerno Italy 2 Department of Logic Nicolaus Copernicus University in Toru´ n Poland TACL 2017 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Outline Inspirations and objectives 1 Half-plane structures 2 Oval structures 3 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Outline Inspirations and objectives 1 Half-plane structures 2 Oval structures 3 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Inspirations and objectives A. N. Whitehead and ovate class of regions 1 Aleksander ´ Sniatycki and half-planes 2 affine geometry 3 follow geometrical intuitions 4 R 2 — «the litmus paper» (regular) open convex subsets of I 5 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Inspirations and objectives A. N. Whitehead and ovate class of regions 1 Aleksander ´ Sniatycki and half-planes 2 affine geometry 3 follow geometrical intuitions 4 R 2 — «the litmus paper» (regular) open convex subsets of I 5 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Inspirations and objectives A. N. Whitehead and ovate class of regions 1 Aleksander ´ Sniatycki and half-planes 2 affine geometry 3 follow geometrical intuitions 4 R 2 — «the litmus paper» (regular) open convex subsets of I 5 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Inspirations and objectives A. N. Whitehead and ovate class of regions 1 Aleksander ´ Sniatycki and half-planes 2 affine geometry 3 follow geometrical intuitions 4 R 2 — «the litmus paper» (regular) open convex subsets of I 5 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Inspirations and objectives A. N. Whitehead and ovate class of regions 1 Aleksander ´ Sniatycki and half-planes 2 affine geometry 3 follow geometrical intuitions 4 R 2 — «the litmus paper» (regular) open convex subsets of I 5 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Inspirations and objectives A. N. Whitehead and ovate class of regions 1 Aleksander ´ Sniatycki and half-planes 2 affine geometry 3 follow geometrical intuitions 4 R 2 — «the litmus paper» (regular) open convex subsets of I 5 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Outline Inspirations and objectives 1 Half-plane structures 2 Oval structures 3 G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Basic notions of ´ Sniatycki’s approach We begin with an examination of triples � R , ≤ , H � in which: R is a non-empty set whose elements are called regions, � R , ≤� is a complete Boolean lattice, H ⊆ R is a set whose elements are called half-planes (we assume that 1 and 0 are not half-planes). G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Specific axioms for half-planes h ∈ H −→ − h ∈ H ( H1 ) � ∃ h ∈ H ∀ i ∈{ 1 , 2 , 3 } ( x i · h � 0 ∧ x i · − h � 0 ) ∨ ∀ x 1 , x 2 , x 3 ∈ R ∃ h 1 , h 2 , h 3 ∈ H ( x 1 ≤ h 1 ∧ x 2 ≤ h 2 ∧ x 3 ≤ h 3 ∧ ( H2 ) � x 1 + x 2 ⊥ h 2 ∧ x 1 + x 3 ⊥ h 2 ∧ x 2 + x 3 ⊥ h 1 ) G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Specific axioms for half-planes h ∈ H −→ − h ∈ H ( H1 ) � ∃ h ∈ H ∀ i ∈{ 1 , 2 , 3 } ( x i · h � 0 ∧ x i · − h � 0 ) ∨ ∀ x 1 , x 2 , x 3 ∈ R ∃ h 1 , h 2 , h 3 ∈ H ( x 1 ≤ h 1 ∧ x 2 ≤ h 2 ∧ x 3 ≤ h 3 ∧ ( H2 ) � x 1 + x 2 ⊥ h 2 ∧ x 1 + x 3 ⊥ h 2 ∧ x 2 + x 3 ⊥ h 1 ) G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Specific axioms for half-planes ∀ h 1 , h 2 , h 3 ∈ H ( h 2 ≤ h 1 ∧ h 3 ≤ h 1 −→ h 2 ≤ h 3 ∨ h 3 ≤ h 2 ) ( H3 ) h 1 h 2 h Figure: In Beltramy-Klein model there are half-planes contained in a given one but incomparable in terms of ≤ . In the picture above h 1 and h 2 are both parts of h , yet neither h 1 ≤ h 2 nor h 2 ≤ h 1 . The purpose of ( H3 ) is to ensure that parallelity of lines is a Euclidean relation. G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Specific axioms for half-planes ∀ h 1 , h 2 , h 3 ∈ H ( h 2 ≤ h 1 ∧ h 3 ≤ h 1 −→ h 2 ≤ h 3 ∨ h 3 ≤ h 2 ) ( H3 ) h 1 h 2 h Figure: In Beltramy-Klein model there are half-planes contained in a given one but incomparable in terms of ≤ . In the picture above h 1 and h 2 are both parts of h , yet neither h 1 ≤ h 2 nor h 2 ≤ h 1 . The purpose of ( H3 ) is to ensure that parallelity of lines is a Euclidean relation. G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Lines and parallelity relation Definition (of a line) L ∈ P ( H ) is a line iff there is a half-plane h such that L = { h , − h } : df L ∈ L ←→ ∃ h ∈ H L = { h , − h } . ( df L ) Definition (of parallelity relation) L 1 , L 2 ∈ L are parallel iff there are half-planes h ∈ L 1 and h ′ ∈ L 2 which are disjoint: df ←→ ∃ h ∈ L 1 ∃ h ′ ∈ L 2 h ⊥ h ′ . L 1 � L 2 ( df � ) In case L 1 and L 2 are not parallel we say they intersect and write: ‘ L 1 ∦ L 2 ’. G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Angles and bowties. . . Definition Given two intersecting lines L 1 and L 2 by an angle we understand a region x such that for h 1 ∈ L 1 and h 2 ∈ L 2 we have x = h 1 · h 2 : df ←→ ∃ L 1 , L 2 ∈ L ( L 1 ∦ L 2 ∧ ∃ h 1 ∈ L 1 ∃ h 2 ∈ L 2 x = h 1 · h 2 ) . x is an angle An angle x is opposite to an angle y iff there are h 1 , h 2 ∈ H such that x = h 1 · h 2 and y = − h 1 · − h 2 . A bowtie is the sum of an angle and its opposite. Notice that every pair L 1 = { h 1 , − h 1 } , L 2 = { h 2 , − h 2 } of non-parallel lines determines exactly four pairwise disjoint angles: h 1 · h 2 , h 1 · − h 2 , − h 1 · h 2 and − h 1 · − h 2 . G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Angles and bowties. . . Definition Given two intersecting lines L 1 and L 2 by an angle we understand a region x such that for h 1 ∈ L 1 and h 2 ∈ L 2 we have x = h 1 · h 2 : df ←→ ∃ L 1 , L 2 ∈ L ( L 1 ∦ L 2 ∧ ∃ h 1 ∈ L 1 ∃ h 2 ∈ L 2 x = h 1 · h 2 ) . x is an angle An angle x is opposite to an angle y iff there are h 1 , h 2 ∈ H such that x = h 1 · h 2 and y = − h 1 · − h 2 . A bowtie is the sum of an angle and its opposite. Notice that every pair L 1 = { h 1 , − h 1 } , L 2 = { h 2 , − h 2 } of non-parallel lines determines exactly four pairwise disjoint angles: h 1 · h 2 , h 1 · − h 2 , − h 1 · h 2 and − h 1 · − h 2 . G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Angles and bowties. . . Definition Given two intersecting lines L 1 and L 2 by an angle we understand a region x such that for h 1 ∈ L 1 and h 2 ∈ L 2 we have x = h 1 · h 2 : df ←→ ∃ L 1 , L 2 ∈ L ( L 1 ∦ L 2 ∧ ∃ h 1 ∈ L 1 ∃ h 2 ∈ L 2 x = h 1 · h 2 ) . x is an angle An angle x is opposite to an angle y iff there are h 1 , h 2 ∈ H such that x = h 1 · h 2 and y = − h 1 · − h 2 . A bowtie is the sum of an angle and its opposite. Notice that every pair L 1 = { h 1 , − h 1 } , L 2 = { h 2 , − h 2 } of non-parallel lines determines exactly four pairwise disjoint angles: h 1 · h 2 , h 1 · − h 2 , − h 1 · h 2 and − h 1 · − h 2 . G. Gerla, R. Gruszczy´ nski Point-free affine geometry
Inspirations and objectives Half-plane structures Oval structures Angles and bowties. . . Definition Given two intersecting lines L 1 and L 2 by an angle we understand a region x such that for h 1 ∈ L 1 and h 2 ∈ L 2 we have x = h 1 · h 2 : df ←→ ∃ L 1 , L 2 ∈ L ( L 1 ∦ L 2 ∧ ∃ h 1 ∈ L 1 ∃ h 2 ∈ L 2 x = h 1 · h 2 ) . x is an angle An angle x is opposite to an angle y iff there are h 1 , h 2 ∈ H such that x = h 1 · h 2 and y = − h 1 · − h 2 . A bowtie is the sum of an angle and its opposite. Notice that every pair L 1 = { h 1 , − h 1 } , L 2 = { h 2 , − h 2 } of non-parallel lines determines exactly four pairwise disjoint angles: h 1 · h 2 , h 1 · − h 2 , − h 1 · h 2 and − h 1 · − h 2 . G. Gerla, R. Gruszczy´ nski Point-free affine geometry
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