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Quasiorders on universal algebras. (English. Russian original) Zbl 0824.08002

Algebra Logic 32, No. 3, 164-173 (1993); translation from Algebra Logika 32, No. 3, 308-325 (1993).
A concept of quasiorder on a universal algebra is investigated. A binary relation \(q\) on the universe \(A\) of an algebra \(\mathfrak A\) is called a quasiorder on \(\mathfrak A\) if 1) \(q\) is a quasiorder on \(A\), 2) for every function \(f(x_ 1,\dots, x_ n)\) in the signature of \(\mathfrak A\) and for arbitrary pairs \(\langle a_ 1, b_ 1\rangle,\dots, \langle a_ n, b_ n\rangle\), the pair \(\langle f(a_ 1,\dots, a_ n), f(b_ 1,\dots, b_ n)\rangle\) belongs to \(q\).
This definition of quasiorders on universal algebras differs from the definitions of partially ordered groups, rings, etc. The set \(\text{Qord }{\mathfrak A}\) of all quasiorders on a universal algebra \(\mathfrak A\) is an algebraic lattice with respect to inclusion. The authors point out many properties of this lattice. Some of them are analogous to properties of \(\text{Con }{\mathfrak A}\).

MSC:

08A30 Subalgebras, congruence relations
06B15 Representation theory of lattices
06F25 Ordered rings, algebras, modules
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References:

[1] A. I. Mal’tsev, ”Toward a general theory of algebraic systems,”Mat. Sb. (New Series),33(77), 3–20 (1954).
[2] G. A. Fraser and R. Horn, ”Congruence relations in a direct product,”Proc. Am. Math. Soc.,26, No.3, 390–394 (1970). · Zbl 0241.08004
[3] G. Czedli and A. Lenkehegli, ”On classes of ordered algebras and quasiorder distributivity,”Acta Sc. Math. (Szeged.),46, No.1–4, 41–54 (1983).
[4] A. G. Pinus,Congruence Modular Varieties of Algebras [in Russian], Irkutsk University, Irkutsk (1986). · Zbl 0714.08003
[5] J. Hagemann and A. Mitschke, ”Onn-permutable congruences,”Alg. Univ.,3, No.1, 8–12 (1973). · Zbl 0273.08001
[6] I. Chajda,Algebraic Theory of Tolerance Relations, Univ. Palackeho Olomouc, Olomouc (1991). · Zbl 0747.08001
[7] J. T. Baldwin and J. A. Berman, ”A model-theoretic approach to Malcev conditions,”J. Symb. Log.,12, No.2, 277–288 (1977). · Zbl 0412.03016
[8] E. Fried, G. Grätzer, and R. Quackenbush, ”Uniform congruence schemes,”Alg. Univ.,10, No.2, 176–189 (1980). · Zbl 0431.08002
[9] A. Day, ”A note on the congruence extension property,”Alg. Univ.,1, No.2, 234–235 (1971). · Zbl 0228.08001
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