Moderately partial algebras whose equivalence relations are congruences
A. V. Reshetnikovab a National Research University of Electronic Technology (Moscow)
b Moscow Center for Fundamental and Applied Mathematics of M. V. Lomonosov Moscow State University (Moscow)
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
$\varphi$ 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
$f(x_1, ..., x_i, ..., x_n) = x_i$).
Keywords:
moderately partial algebra, partial infinite-ary groupoid, congruence lattice, equivalence relation lattice.
UDC:
512.548.2 + 512.579
Received: 21.12.2020
Accepted: 21.02.2021
DOI:
10.22405/2226-8383-2018-22-1-292-303