Axiomatizability of the Class of Subdirectly Irreducible Acts over an Abelian Group
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Algebra and Logic, Vol. 59, No. 5, November, 2020 (Russian Original Vol. 59, No. 5, September-October, 2020)
AXIOMATIZABILITY OF THE CLASS OF SUBDIRECTLY IRREDUCIBLE ACTS OVER AN ABELIAN GROUP A. A. Stepanova1∗ and D. O. Ptakhov2∗∗
UDC 510.67:512.54
Keywords: axiomatizable class of algebras, act over group, subdirectly irreducible act over group. Abelian groups are described over which the class of all subdirectly irreducible acts is axiomatizable. Also some properties of subdirectly irreducible acts over Abelian groups are studied. It is proved that all connected acts over an Abelian group are subdirectly irreducible iff the group is totally ordered. A description of monoids over which some class of acts is axiomatizable is a standard problem in the model theory of acts. For the classes of flat acts, projective acts, and free acts, this problem was solved in [1-3]. For the class of regular acts, such a description was obtained in [4, 5]. In the present paper, we give a description of Abelian groups G over which the class of all subdirectly irreducible acts is axiomatizable. Furthermore, we study properties of subdirectly irreducible acts over Abelian groups. It is proved that all connected acts over an Abelian group are subdirectly irreducible iff the group is totally ordered. 1. PRELIMINARY INFORMATION Let A be an algebra. Denote by Con(A) the set of all congruences on A, and by 0A the zero congruence of A. An algebra A is said to be subdirectly irreducible if it contains a least congruence other that 0A. Clearly, A is subdirectly irreducible iff {θ ∈ Con(A) | θ = 0A} = 0A. We recall some notions from the theory of acts [6]. ∗
Supported by RF Ministry of Education and Science (Suppl. Agreement No. 075-02-2020-1482-1 of 21.04.2020). Supported by RFBR, project No. 17-01-00531.
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1
School of Natural Sciences, Far Eastern Federal University, Vladivostok, Russia; [email protected]. School of Natural Sciences, Far Eastern Federal University, Vladivostok, Russia; [email protected]. Translated from Algebra i Logika, Vol. 59, No. 5, pp. 582-593, September-October, 2020. Russian DOI: 10.33048/alglog.2020.59.505. Original article submitted March 3, 2019; accepted November 27, 2020. 2
c 2020 Springer Science+Business Media, LLC 0002-5232/20/5905-0395
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Let S be a monoid. A left S-act (or an act over S, or simply an act) S A is a nonempty set A on which the action of the monoid S is defined, with the identity element of S acting on A identically. Elements a, b ∈ A are said to be connected in S A if there exist n ∈ ω, ci ∈ A (0 i n), and sj , tj ∈ S (1 j n) such that a = c0 , b = cn , and si ci−1 = ti ci for any i, 1 i n. An act S A is Ai of acts S Ai (i ∈ I) is their connected if any two of its elements are connected. The coproduct i∈ I
disjoint union. A congruence θ on an act S A is an equivalence relation on a set A such that (a, b) ∈ θ ⇒ (sa, sb) ∈ θ for any a, b ∈ A and any s ∈ S. Let S B be a subact of S A and θ be a congruence on S A. Denote by ρ(B) the Riesz congruence on S A, i.e., (a, b) ∈ ρ(
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