The variety of all commutative BCK-algebras is generated by its finite members as a quasivariety

Abstract

Abstract. We prove the result announced by the title as well as some of its consequences.It is well known that the variety of Łukasiewicz algebras is generated by its finite members (see e.g. [20]). W. Blok and I. Ferreirim [3] proved that the variety of all Łukasiewicz algebras is generated by its finite members as a quasi-variety. In this paper we show that this is also the case for the variety of all commutative BCK-algebras as well as some of its subvarieties. It is worth to note that unlike the case of subvarieties of the variety of all Łukasiewicz algebras, there are subvarieties of the variety of all commutative BCK-algebras which are not generated by their finite members [29,38].The main result of the paper was obtained in early '90s. Since then there were many papers on BCK-algebras, BCK-algebras with condition (S) (pocrims) and bounded commutative BCK-algebras (Wajsberg algebras, MV-algebras). Let me list some important papers suggested by an anonymous referee.• A good study of the varieties of BCK-algebras (including some aspects of commutative BCK-algebras) was done by W. Blok and J. Raftery in [5]. The references given in this paper provide a fairly comprehensive vision of the status of BCK-algebras until 1995. The implicative presentation of BCK-algebras tends to be used today, because they are the implicational subreduct of commutative integral residuated lattices.• BCK-algebras with condition S are currently known as pocrims: partially ordered, commutative, residuated, integral monoids. The varieties of were also studied by W. Blok and J. Raftery in [6]. This paper contains a complete bibliography on pocrims and BCK-algebras.• Partially naturally ordered commutative monoids with residuation are a special case of pocrims known as hoops, and they are studied in [4,2,3].• Bounded Commutative BCK-algebras, in their implicational presentation, are also known as CN-algebras [20] and Wajsberg algebras [15]. In fact, they are term-wise equivalent to Chang's MV-algebras introduced in [9] (see also [20, 15, 23] and [11]). MV-algebras can be seen as the unit segment of commutative lattice ordered groups with a strong unit (see [10] and [20] for and [24] for the general case).• The literature on MV-algebras is very large. Since Mundici's work [23], in which he studies the relationship between the many valued Łukasiewicz logic and AF C*-algebras, using the categorical equivalence between MValgebras and commutative ℓ-groups with strong unit, the theory of the MV-algebras has developed a great deal in different directions. A general approach to the theory of MV-algebras can be found in the book [11] and the paper [16]. The lattice of all subvarieties of MV-algebras was studied in [20], and a complete equational presentation of these subvarieties can be found in [13].

On Solutions to the Set-Theoretical Yang-Baxter Equation in Wajsberg-Algebras

“…Moreover, bounded commutative BCI/BCK-algebras (BCK-algebras are special cases of BCI-algebras, for example, see [21]) , in their implicational notation, are known as Wajsberg algebras (see also [22]). …”