## 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

### Keywords:

quasiorder; algebraic lattice
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### References:

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