Axiomatizability of the class of subdirectly irreducible acts over a commutative monoid

An act ${}_SA$ is called subdirectly irreducible if
$$\bigcap\{\rho_i\mid \rho_i\neq\Delta, i\in I\}\neq\Delta$$
for every family of congruences $\rho_i$ on ${}_SA$ ($i\in I$) where $\Delta$ is zero congruence on ${}_SA$. The interest in the study of such acts is caused by Birkhoff's theorem, according to which any algebra is isomorphic to a subdirect product of subdirectly irreducible algebras [1]. The question of axiomatizability of the class of subdirectly irreducible acts over an Abelian group was studied by Stepanova A.A. and Ptakhov D.O. [2]. We describe commutative monoids, the class of subdirectly irreducible acts over which is axiomatizable.
Supported by RF Ministry of Education and Science (Suppl. Agreement No. 075-02-2021-1395 of 01.06.2021).

Список литературы

1. G. Birkhoff, “Subdirect unions in universal algebra”, Bulletin of the American Mathematical Society, 50 (1944), 764–768
2. А. А. Степанова, Д. О. Птахов, “Аксиоматизируемость класса подпрямо неразложимых полигонов над абелевой группой”, Алгебра и логика, 59:5 (2020), 582–593    ; Algebra and Logic, 59:5 (2020), 395–403