Quasiorders on universal algebras
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 \(\text{Con }{\mathfrak A}\).
scientific article; zbMATH DE number 738951