Moderately partial algebras whose equivalence relations are congruences

A. V. Reshetnikov

Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 292-303

Résumé

Consider partial algebras whose equivalence relations are congruences. The problem of description of such partial algebras can be reduced to the problem of description of partial $n$-ary groupoids with the similar condition. In this paper a concept of moderately partial operation is used. A description is given for the moderately partial operations preserving any equivalence relation on a fixed set. Let $A$ be a non-empty set, $f$ be a moderately partial operation, defined on $A$ (i.e. if we fix all of the arguments of $f$, except one of them, we obtain a new partial operation $\varphi$ such that its domain $\mathrm{dom}\, \varphi$ satisfies the condition $|\mathrm{dom}\, \varphi| \ge 3$). Let any equivalence relation on the set $A$ be stable relative to $f$ (in the other words, the congruence lattice of the partial algebra $(A,\{f\})$ coinsides the equivalence relation lattice on the set $A$). In this paper we prove that in this case the partial operation $f$ can be extended to a full operation $g$, also defined on the set $A$, such that $g$ preserves any equivalence relation on $A$ too. Moreover, if the arity of the partial operation $f$ is finite, then either $f$ is a partial constant (i.e. $f(x) = f(y)$ for all $x,y \in \mathrm{dom}\, f$), or $f$ is a partial projection (there is an index $i$ such that all of the tuples $x = (x_1, ..., x_n) \in \mathrm{dom}\, f$ satisfy the condition $f(x_1, ..., x_i, ..., x_n) = x_i$).

Détail

Keywords: moderately partial algebra, partial infinite-ary groupoid, congruence lattice, equivalence relation lattice.