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On representation of cyclically ordered sets

1989; Springer Nature; Volume: 39; Issue: 1 Linguagem: Inglês

10.21136/cmj.1989.102284

1572-9141

Vítězslav Novák, Miroslav Novotný,

Rough Sets and Fuzzy Logic

In [5] we have constructed, for any cardinal m, an m-universal cyclically ordered set.The m-universality is meant there in the following sense: For any cyclically ordered set G with cardinality ^ m there exists a subset G' of the universal set con structed such that G is a strong homomorphic image of G'.Here we present a con struction of a set with an asymmetric and cyclic ternary relation such that any cyclically ordered set of cardinality % m is isomorphic with its suitable subset. POWER OF TERNARY STRUCTURESLet G be a set and C a ternary relation on G.The pair G = (G, C) will be called a ternary structure.Sometimes we denote by %>(G) the carrier of this structure, i.e.'#(G) = G, and by at(G) the relation ofthis structure, i.e. 0t{G) = C.A ternary structure G = (G, C) is called reflexive, iff x, y, z e G, card {x, y, z] ^ 2 => (x, y, z) e C; irreflexive, iff x, y, z e G, card {x, y, z} ^ 2 => (x, y, z) є C; symmetric, iff x, j, z e G, (x, j, z) є C => (z, y, x) є С; asymmetric, iff x, j, z є G, (x, y, z) є C => (z, y, x) є C; cyclic, iff x, y, z є G, (x, y, z) є C => (j, z, x) є C; transitive, iff x, y, z, и є G, (x, y, z) є C, (x, z, w) є C => (x, y, w) є C.A cyclically ordered set is a ternary structure which is asymmetric, cyclic and transitive.A cycle is a cyclically ordered set G = (G, C) which is complete, i.e. x 3 y, z є G, x ф у ф z Ф x => (x, y, z) є C or (z, y, x) e C.Let G = (G, C) be a ternary structure and H Я G.We call the subset H discrete, iff Я 3 n С = 0.An element x є G will be called isolated, iff {x, y, z] is a discrete subset of G for any y e G, z e G.A direct sum, direct product and a homomorphism of ternary structures are defined in the obvious way.By the symbol Hom (G, H) we denote the set of all homomorphisms of G into H.An isomorphism of G onto H is a bijective homo morphism / of G onto H such that/ -1 is a homomorphism of H onto G.An injective homomorphism / of G into # such that f~l is a homomorphism off(G) onto G 1 will be called an embedding.