BEYOND BARYCENTRIC ALGEBRAS AND CONVEX SETS A. KOMOROWSKI, A. B. - PDF document

BEYOND BARYCENTRIC ALGEBRAS AND CONVEX SETS A. KOMOROWSKI, A. B. ROMANOWSKA Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warsaw, Poland J. D. H. SMITH Department of Mathematics, Iowa State University, Ames, Iowa, 50011, USA 1

OUTLINE • Affine spaces, convex sets and barycentric algebras • Extended barycentric algebras • q -convex sets and q -barycentric algebras • Threshold barycentric algebras and threshold affine spaces 2

AFFINE SUBSPACES of R n R - the field of reals; I ◦ :=]0 , 1[= (0 , 1) ⊂ R . The line L x,y through x, y ∈ R n : L x,y = { xy p = x (1 − p ) + yp ∈ R n | p ∈ R } . A ⊆ R n is a (non-trivial) affine subspace of R n if, together with any two distinct points x and y , it contains the line L x,y . One obtains an algebra ( A, { p | p ∈ R } ). 3

CONVEX SUBSETS of R n The line segment I x,y joining the points x, y : I x,y = { xy p = x (1 − p ) + yp ∈ R n | p ∈ I ◦ } . C ⊆ R n is a (non-trivial) convex subset of R n if, together with any two distinct points x and y , it contains the line segment I x,y . One obtains an algebra ( C, { p | p ∈ I ◦ } ). 4

AFFINE SPACES F - a subfield of R An affine space over F (or affine F -space ) - an algebra ( A, F ) , where F = { p | p ∈ F } and xyp = p ( x, y ) = x (1 − p ) + yp. Note : ( A, F ) is equivalent to the algebra n n � � � � � � A, x i r i r i = 1 in F . � � i =1 i =1 5

THE VARIETY OF AFFINE F -SPACES THEOREM: The class F of all affine F -spaces is a variety (equationally-defined class). It is axiomatized by the following: idempotence : xxp = x, entropicity : xyp ztp q = xzq ytq p, affine identities : xyp xyq r = xy pqr, trivial identities : xy 0 = x = yx 1, for all p, q, r ∈ F . For each p ∈ F with p � = 0 , 1, the reduct ( A, p ) of an affine F -space ( A, F ) is an (idempotent and entropic) quasigroup. Hence the algebra ( A, F \ { 0 , 1 } ) is an (idempotent and entropic) multi-quasigroup . 6

BARYCENTRIC ALGEBRAS F - a subfield of R , I ◦ :=]0 , 1[= (0 , 1) ⊂ F . A convex set over F (or convex F -set ) - an algebra ( A, I o ) , where I o = { p | p ∈ I ◦ } . The class C of convex F -sets forms a quasivariety. The variety generated by C is the variety B of barycentric algebras . The variety B is axiomatized by the following: idempotence (I): xxp = x , skew-commutativity (SC): xy p = xy 1 − p =: xy p ′ , skew-associativity (SA): [ xyp ] z q = x [ yz q/ ( p ◦ q ) ] p ◦ q for all p, q ∈ I ◦ , where p ◦ q = ( p ′ q ′ ) ′ = p + q − pq . 7

EXAMPLES • Convex subsets of affine F -spaces under the operations xy p = xp ′ + yp = x (1 − p ) + yp for each p ∈ I ◦ . The subquasivariety C of the variety B is defined by the cancellation laws ( xy p = xz p ) → ( y = z ) for all operations p of I ◦ . • “Stammered” semilattices ( S, · ) with the operation x · y = xyp for all p ∈ I ◦ . They form the subvariety SL of B defined by xy p = xy r for all p, r ∈ I ◦ . • Certain sums of convex sets over semilattices. THEOREM: Each barycentric algebra is a subalgebra of a P� lonka sum of convex sets over its semilattice replica. 8

EXTENDED BARYCENTRIC ALGEBRAS Barycentric algebras may be considered as extended barycentric algebras ( A, I ), where I = [0 , 1] ⊂ R , and with the operations 0 and 1 defined by xy 0 = x and xy 1 = y. Skew associativity may also be written as: [ xyp ] zq = x [ yz ( p ◦ q → q ) ] p ◦ q (SA), where  1 if p = 0;  p → q = q/p otherwise  Proposition: The class B of extended barycentric algebras is a variety, specified by the identities (I), (SC), (SA) and the two above. 9

EXTENDING THE CONCEPTS of a CONVEX SET and of a BARYCENTRIC ALGEBRA Want to extend the concepts of a convex set and a barycentric algebra, while retaining as many key properties of barycentric algebras as possible. Two types of extensions obtained by: 1. using different intervals of the field F ; 2. using more general rings. 10

q -CONVEX SETS A convex subset C of an affine F -space A is an I ◦ -subreduct of A (subalgebra of ( A, I ◦ )). Replace the interval I ◦ by an open interval ] q, q ′ [, where q ∈ F with q ≤ 1 / 2 and q ′ = 1 − q . � � � � C, ] q, q ′ [ A, ] q, q ′ [ A subalgebra of is called a q-convex set . The class C q of all q -convex subsets of affine F -spaces is a quasivariety. The variety B q generated by the quasivariety C q is called the variety of q-barycentric algebras . Note: C 0 = C , and B 0 = B . B 1 / 2 = CBM (the variety of commutative binary modes) 11

SOME BASIC PROPERTIES Proposition: Let t ∈ F . If −∞ < t < 0, then under the operations of [ t, t ′ ] the line F is generated by { 0 , 1 } . If 0 < t < 1 / 2, then under the operations of [ t, t ′ ] the interval I is generated by { 0 , 1 } . Proposition: Free B q -algebra over X is isomorphic to the subalgebra generated by X in the ] q, q ′ [-reduct ( XF, ] q, q ′ [) of the free affine F -space ( XF, F ) over X . Corollary: In each q-convex set, the operations of [ t, t ′ ], for t � = 0 and t � = 1 / 2, either generate all operations of I ◦ , or all operations of F . 12

CLASSIFICATION THEOREM Let q ∈ F with q ≤ 1 / 2. Then each variety B q is equivalent to one of the following: (a) the variety CBM of commutative binary modes, if q = 1 / 2 ; (b) the variety B of barycentric algebras, if 0 ≤ q < 1 / 2 ; (c) the variety A of affine F -spaces, if q < 0. 13

THRESHOLD ALGEBRAS Set a threshold t , where t = −∞ or t ∈ F with t ≤ 1 / 2. For elements x, y of an affine F -space, define  x if r < t ;    if t ≤ r ≤ t ′ ; xy r = xy r = x (1 − r ) + yr  if r > t ′  y  for r ∈ F . Then the binary operations r are described as threshold- t affine combinations ( small, moderate and large respectively). For a given threshold t , the algebra ( A, F ), where F = { r | r ∈ F } , is called a threshold- t affine F -space . 14

Proposition: Let t be a threshold. Let A be an affine F -space. Then under the threshold- t affine combinations r for r ∈ F , the threshold- t affine F -space ( A, F ) is idempotent, entropic and skew-commutative. For a given threshold t , the class A t of threshold- t affine F -spaces is the variety generated by the class of affine F -spaces under the threshold- t affine combinations. For 0 ≤ t ≤ 1 / 2, similar definitions provide the concepts of threshold- t convex combinations , threshold- t convex sets , and the variety B t of threshold- t barycentric algebras . If t = −∞ , then A = A −∞ . If t = 1 / 2, then A 1 / 2 ≃ B 1 / 2 ≃ CBM , If 0 < t < 1 / 2, then A t ≃ B t ≃ B , A 0 ≃ B . 15

MAIN RESULT THEOREM Each variety of threshold affine F -spaces is equivalent to one of the following classes: (a) the variety A of affine F -spaces; (b) the variety B of extended barycentric algebras; (c) the variety CBM of extended commutatve binary modes. 16

Some references • Gudder, S.P.: Convex structures and operational quantum mechanics, Comm. Math. Phys. 29 (1973), 249–264. • Jeˇ zek, J., Kepka, T.: The lattice of varieties of commutative abelian distributive groupoids, Algebra Universalis 5 (1975), 225–237. • Komorowski, A., Romanowska, A., Smith, J.D.H.: Keimel’s problem on the algebraic axiomatization of convexity, Algebra Universalis (2018), 79-22. • Komorowski, A., Romanowska, A., Smith, J.D.H.: Barycentric algebras and beyond, Algebra Universalis (2019), 80:20. • Neumann, W.D.: On the quasivariety of convex subsets of affine spaces, Arch. Math. (Basel) 21 (1970), 11–16. 17

• Or� lowska, E., Romanowska, A.B., Smith, J.D.H.: Abstract barycentric algebras, Fund. Informaticae 81 (2007), 257–273. • Ostermann, F., Schmidt, J.: Der baryzentrische Kalk¨ ul als axiomatische Grundlage der affinen Geometrie, J. Reine Angew. Math. 224 (1966), 44–57. • Romanowska, A.B., Smith, J.D.H.: Modal Theory, Heldermann, Berlin, 1985. • Romanowska, A.B., Smith, J.D.H.: On the structure of barycentric algebras, Houston J.Math. 16 (1990), 431–448. • Romanowska, A.B., Smith, J.D.H.: Modes, World Scientific, Singapore, 2002. • Skornyakov, L.A.: Stochastic algebras, Izv. Vyssh. Uchebn. Zaved. Mat. 29 (1985), 3–11.