Quasi Variety Research Articles

A Lattice of Quasivarieties of Normal Cryptogroups

A normal cryptogroup S is a completely regular semigroup in which ${\cal H}$ is a congruence and $S/{{\cal H}}$ is a normal band. We represent S as a strong semilattice of completely simple semigroups, and may set $S = [Y; S_\alpha , \chi_{\alpha ,\beta }].$ For each $\alpha \in Y,$ we set $S_\alpha = {\cal M}(I_\alpha , G_\alpha , \Lambda _\alpha ; P_\alpha),$ and represent $\chi_{\alpha ,\beta }$ by means of an h-quintuple $(\xi _{\alpha ,\beta }, - , \omega _{\alpha ,\beta}, - , \eta_{\alpha ,\beta }).$ These parameters are used to characterize certain quasivarieties of normal cryptogroups. Specifically, we construct the lattice of quasivarieties generated by the (quasi)varieties ${\cal L}{\cal Z}, {\cal S}, {\cal R}{\cal Z}, {\cal L}{\cal N}{\cal B}, {\cal R}{\cal N}{\cal B}, {\cal G}$ and ${\cal C}{\cal S}.$ This is the lattice generated by the lattice of quasivarieties of normal bands, groups and completely simple semigroups. We also determine the B-relation on the lattice of all quasivarieties of normal cryptogroups. Each quasivariety studied is characterized in several ways.

Notions of Discrimination

As an outgrowth of the study of algebraic geometry over groups, discriminating groups were introduced. Many important universal type groups such as Higman's universal group and Thompson's group F were shown to be discriminating. Squarelike groups were then introduced to better capture axiomatic properties of discrimination. In the present article squarelike groups are reinterpreted in terms of discrimination of quasi varieties, and the relationship with an older version of discrimination, termed varietal discrimination here, is studied.

Functional Completeness and Axiomatizability within Belnap's Four-Valued Logic and its Expansions

ABSTRACT In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. Further, finding axiomatizations of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic.

Higman's Embedding Theorem in a General Setting and Its Application to Existentially Closed Algebras

For a quasi variety of algebras K, the Higman Theorem is said to be true if every recursively presented K-algebra is embeddable into a finitely presented K-algebra; the Generalized Higman Theorem is said to be true if any K-algebra which is recursively presented over its finitely generated subalgebra is embeddable into a K-algebra which is finitely presented over this subalgebra. We suggest certain general conditions on K under which (1) the Higman Theorem implies the Generalized Higman Theorem; (2) a finitely generated K-algebra A is embeddable into every existentially closed K-algebra containing a finitely generated K-algebra B if and only if the word problem for A is Q-reducible to the word problem for B. The quasi varieties of groups, torsion-free groups, and semigroups satisfy these conditions.

P-varieties - a signature independent characterization of varieties of ordered algebras

This paper is concerned mainly with classes (categories) of ordered algebras which in some signature are axiomatizable by a set of inequations between terms (‘varieties’ of ordered algebras) and also classes which are axiomatizable by implications between inequations (‘quasi varieties’ of ordered algebras). For example, if the signature contains a binary operation symbol (for the monoid operation) and a constant symbol (for the identity) the class of ordered monoids M can be axiomatized by a set of inequations (i.e. expressions of the form t≤ t'. However, if the signature contains only the binary operation symbol, the same class M cannot be so axiomatized (since it is not now closed under subalgebras). Thus, there is a need to find structural, signature independent conditions on a class of ordered algebras which are necessary and sufficient to guarantee the existence of a signature in which the class is axiomatizable by a set of inequations (between terms in this signature). In this paper such conditions are found by utilizing the notion of ‘P-categories’. A P-category C is a category such that each ‘Hom-set’ C( a,b) is equipped with a distiguished partial order which is preserved by composition. Aside from proving the characterization theorem, it is also the purpose of the paper to begin the investigation of P-categories.