Generalized Quasiorders and the Galois Connection End-gQuord

TL;DR

This paper introduces generalized quasiorders, extending the concept of quasiorders, to characterize algebras with clone of term operations determined by translations, generalizing affine complete algebras.

Contribution

It defines generalized quasiorders and characterizes u-closed monoids as Galois closures of the End-gQuord connection, broadening the understanding of algebraic structures.

Findings

Characterization of generalized quasiorders of arbitrary arity.

Explicit description of minimal u-closed monoids.

Generalization of affine complete algebras.

Abstract

Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) $\rho$ have the property that an $n$-ary operation $f$ preserves $\rho$, i.e., $f$ is a polymorphism of $\rho$, if and only if each translation (i.e., unary polynomial function obtained from $f$ by substituting constants) preserves $\rho$, i.e., it is an endomorphism of $\rho$. We introduce a wider class of relations -- called generalized quasiorders -- of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection End-gQuord, i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly.