Moderately partial algebras whose equivalence relations are congruences

Abstract

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 𝜙 such that its domain dom 𝜙 satisfies the condition |dom 𝜙| > 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 ∈ dom 𝑓), or 𝑓 is a partial projection (there is an index 𝑖 such that all of the tuples 𝑥 = (𝑥1, ..., 𝑥𝑛) ∈ dom 𝑓 satisfy the condition 𝑓(𝑥1, ..., 𝑥𝑖, ..., 𝑥𝑛) = 𝑥𝑖)