Concerning axiomatizability of the quasivariety generated by a finite Heyting or topological Boolean algebra
✍ Scribed by Wlesław Dziobiak
- Publisher
- Springer Netherlands
- Year
- 1982
- Tongue
- English
- Weight
- 852 KB
- Volume
- 41
- Category
- Article
- ISSN
- 0039-3215
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✦ Synopsis
In classes of algebras such as lattices, groups, and rings, there arefinite algebras which individually generate quasivarieties which are not finitely axiomatiza.ble (see [2], [3], [8]). We show here that this kind of algebras also exist in Heyting algebras as well as in topological Boolean algebras. Moreover, we show that the lattiee join of two finitely axiomatizable quasivarieties, each generated by a finite Eeyting: or topological Boolean algebra, respectively, need not be finitely axiomatizable. Finally, 9 we solve problem 4 a.sked in Rautenberg [10].
Preliminaries
A quasivariety (finitely axiomatizable q~asivariety) is the class of all:
algebraic structures of some fixed type which are models of some set (finite set) of qua siidentitics. Equivalently (see [7]), it is a class ci algebraic structures eloscd under the operations of forming: ultraproducts --P,,~. dh'ect products --P, substructures --S and isomorphisms --I. By theqnasivariety generated by a class K of similar algebraic structures, in symbols q(K), we mean the least quasivariety containing" K. It is known ~hat q(lff) = ISP.P~(K) (see [5]). In particular, if K is a finite set of finite algebraic structures then q(K) = ISP(K).
A Heyting algebra (= pseudo-Boolean algebra, see [9]) is an algebra.