Equational Compactness of G-sheaves

Purity and equational compactness play a role at least in the Theories of Modules, Acts, Model, and Category. Adámek and Rosický have studied them categorically, Rothmaler model-theoretically, and some authors, including Banaschewski, Gould, and Normak have studied these notions on G-acts. We take both the group G and the set A in the definition of a G-act to be sheaves and study equationally compact G-sheaves. We get different kinds of equationally compact G-sheaves, study them and their interrelations, give some conditions for their proper behavior, and generalize some of the existing results.     

Equational theory of continuous lattices

It is known that continuous lattices and their morphisms can be obtained as algebras and the corresponding homomorphisms respectively for an (infinitary) algebraic theory. In this paper we present one such theory explicitly.     

Equational aspects of ultrafilter convergence

Continuous maps may be defined as those preserving ultrafilter convergence, but many other relations are preserved by all continuous maps. Sets of equations in such relations define classes of spaces such as T1, T2, T3, compact and completely regular. Topological spaces are characterized by a canonical set of inequalities among ultrafilter convergence relations which, when tightened to equalities, specialize to the compact T1 spaces. The research reported in this paper was supported in part by the National Science Foundation under Grant No. MCS 76-84477. Presented by F. E. J. Linton.     

Equational theory of idempotent algebras

A finitely based equational class of idempotent algebras of type <m, n>,m, n≥2, is two-based. More generally, any finitely based equational class of idempotent algebras of type <m ₁, ..., m_k> withm _i≥2 andk≥2 isk-based.     

Equational treatment of first-order logic

Presented by G. Grätzer.     

Equational classes generated by finite algebras

Algebra universalis -     

Equational spectrum of Hilbert varieties

We prove that an equational class of Hilbert algebras cannot be defined by a single equation. In particular Hilbert algebras and implication algebras are not one-based. Also, we use a seminal theorem of Alfred Tarski in equational logic to characterize the set of cardinalities of all finite irredundant bases of the varieties of Hilbert algebras, implication algebras and commutative BCK algebras: all these varieties can be defined by independent bases of n elements, for each n > 1.      

Equational Noetherianness of rigid soluble groups

A group G is said to be rigid if it contains a normal series of the form G = G ₁ > G ₂ > … > G _m  > G _m + 1 = 1, whose quotients G _i /G _i + 1 are Abelian and are torsion free as right Z[G/G _i ]-modules. In studying properties of such groups, it was shown, in particular, that the above series is defined by the group uniquely. It is known that finitely generated rigid groups are equationally Noetherian: i.e., for any n, every system of equations in x ₁, …, x _n over a given group is equivalent to some of its finite subsystems. This fact is equivalent to the Zariski topology being Noetherian on G ⁿ , which allowed the dimension theory in algebraic geometry over finitely generated rigid groups to have been constructed. It is proved that every rigid group is equationally Noetherian. Supported by RFBR (project No. 09-01-00099) and by the Russian Ministry of Education through the Analytical Departmental Target Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project No. 2.1.1.419). Translated from Algebra i Logika, Vol. 48, No. 2, pp. 258–279, March–April, 2009.     

Equational fragments of systems for arithmetic

We investigate the equational fragments of formal systems for arithmetic by means of the equational theory of f-rings and of their positive cones, starting from the observation that a model of arithmetic is the positive cone of a discretely ordered ring. A consequence of the discreteness of the order is the presence of a discriminator, which allows us to derive many properties of the models of our equational theories. For example, the spectral topology of discrete f-rings is a Stone topology. We also characterize the equational fragment of Iopen, and we obtain an equational version of G?del's First Incompleteness Theorem. Finally, we prove that the lattice of subvarieties of the variety of discrete f-rings is uncountable, and that the lattice of filters of the countably generated distributive free lattice can be embedded into it. Received April 17, 1998; accepted in final form January 23, 2001.     

可换BCK代数的方程系统

In this paper we give three equational systems of commutative BCK-algebra. The first system CBKⅠconsists of the following three identities: ((x·y)·(x·z))·(z·y)=0,x·0=x,x·(x·y)=y·(y·x).The second system CBK Ⅱ consists of the following three identities:0·x=0,x·0=x,(x·y)·(x·z)=(z·y)·(z·x).The third system CBKⅢ consists of the following three identities: x·x=0,x·0=x,(x·y)·(x·z)=(z·y)·(z·x).     

Hybrid Identities and Hybrid Equational Logic

Hybrid identities are sentences in a special second order language with identity. The model classes of sets of hybrid identities are called hybrid solid varieties. We give a Birkhoff-type-characterization of hybrid solid varieties and develop a hybrid equational logic.     

An Equational Characterization of Quasi-injective Modules

Injective modules are known to be precisely those over which every compatible system of linear equations is solvable. In this note, we introduce the concept of strong compatibility to give an analogous, but not generally known characterization of quasi-injectivity.