Cyclically ordered sets

Abstract

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Algebras assigned to ternary relations

“…If L is a distributive lattice then the algebra .LI m/ is called the median algebra derived from L. Of course, there exist median algebras which are not derived from a lattice, see [1] for details, but in every median algebra there can be introduced a ternary relation "between" by putting In what follows, we show how this construction can be generalized and we get a characterization of some important properties of ternary relations by means of identities of their assigned operations. We can consider a number of properties of ternary relations which are used in [1][2][3][4][5][6][7][8][9][10][11] for "betweenness" and for "cyclic orders". .fag Z T .a; b/ fbg/:…”

“…However, there exist also other useful ternary relations for which a similar construction is not already derived, in particular the so-called cyclic orders, see e.g. [4,7,8] and [9].…”

On a power of cyclically ordered sets

“…S, i.e. there exist/eU(x), g e U(y), heU(z) with (/, g, h) e R. Thus dom/ = dom g = dom h = X, where -Y = (X, D n X 3 ) e e <£(//), x, >, z e X and (/(t), g((), h(t)) e C for any (e X. Further/(x) = co = g(>>) = = h(z). By Lemma 5, we have x # >> # z + x.…”

“…) is a cylically ordered set and H c G is such a subset that the induced cyclic order C n H 3 is linear on H, then H = (if, C n if 3 ) is called a cycle in G.…”

Completion of a cyclically ordered group

“…A generahzation of this notion was investigated in a series of papers by Y. Novak and M. Novotny (cf., e.g., [8], [9]; cf. also [6] and the papers quoted there).…”