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 \(\text{Con }{\mathfrak A}\).


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