On the variety generated by Murskii's algebra

It is shown that the variety generated by Murskii's algebra contains uncountably many subvarieties.Presented by K. A. Baker.     

The variety generated by equivalence algebras

Every equivalence relation can be made into a groupoid with the same underlying set if we define the multiplication as follows: xy = x if x,y are related; otherwise, xy = y. The groupoids, obtained in this way, are called equivalence algebras. We find a finite base for the equations of equivalence algebras. The base consists of equations in four variables, and we prove that there is no base consisting of equations in three variables only. We also prove that all subdirectly irreducibles in the variety generated by equivalence algebras are embeddable into the three-element equivalence algebra, corresponding to the equivalence with two blocks on three elements. Received September 21, 1998; accepted in final form May 11, 1999.     

The variety generated by planar modular lattices

We investigate the variety generated by the class of planar modular lattices. The main result is a structure theorem describing the subdirectly irreducible members of this variety.     

一类平坦半环生成的簇

平坦半环是一类重要的加法幂等元半环,它在半环簇理论的研究中扮演着重要的角色.主要研究了次直不可约的平坦半环,以及一类平坦半环生成的簇.给出了次直不可约的nil-平坦半环的等价刻画,证明了当n小于4时,平坦半环S(x₁x₂···x_n)均是有限基底的.     

On the variety generated by all nilpotent lattice-ordered groups

In 1974, J. Martinez introduced the variety of weakly Abelian lattice-ordered groups; it is defined by the identity

     

A representation of homology theory by normal cycles

We prove that the functor associated to the manifold class N{mathcal{N}} of connected normal cycles agrees with the classic homology functor.     

A variety generated by a finite algebra with 2x 0 subvarieties

As was indicated to me by Prof. A. Wroński, the following problem was suggested by Prof. B. Jónsson: is every subvariety of the variety of a finite algebra generated by a finite algebra? In this paper we solve this problem in the negative by constructing a finite algebra that generates a variety having 2^x ₀ subvarieties.     

Completely regular semigroups in the variety generated by the bicyclic semigroup

We shall show that a completely regular semigroup is in the semigroup variety generated by the bicyclic semigroup if and only if it is an orthogroup whose maximal subgroups are abelian. Therefore the lattice of subvarieties of the variety generated by the bicyclic semigroup contains as a sublattice a countably infinite distributive lattice of semigroup varieties, each of which consists of orthogroups with maximal subgroups that are torsion abelian groups. In particular, every band divides a power of the bicyclic semigroup.Presented by B. M. Schein.     

A comprehensive theory of representation for mathematics education

Representation is a difficult concept. Behaviorists wanted to get rid of it; many researchers prefer other terms like “conception” or “reasoning” or even “encoding;” and many cognitive science resarchers have tried to avoid the problem by reducing thinking to production rules.There are at least two simple and naive reasons for considering representation as an important subject for scientific study. The first one is that we all experience representation as a stream of internal images, gestures and words. The second one is that the words and symbols we use to communicate do not refer directly to reality but to represented entities: objects, properties, relationships, processes, actions, and constructs, about which there is no automatic agreement between two persons. It is the purpose of this paper to analyse this problem, and to try to connect it with an original analysis of the role of action in representation. The issue is important for mathematics education and even for the epistemology of mathematics, as mathematical concepts have their first roots in the action on, and in the representation of, the physical and social world; even though there may be a great distance today between that pragmatical and empirical source, and the sophisticated concepts of contemporary mathematics.     

A representation theory for the impulse control of jump processes

A state space representation theory for the impulse control of a quite general class of non-Markov jump processes is developed. Control decisions are based upon observations of past histories of an input jump process and observations of the current state of the corresponding controlled output. A verification theorem establishes that a solution of a system of quasi variational inequalities gives rise naturally to an optimal impulse policy. The proof of optimality relies upon an extended version of the well-known Dynkin formula for Markov processes.     

Model theoretic properties in the variety generated by a primal algebra

We show that the model theoretic and syntactic properties of the structures in the variety generated by a primal algebra are essentially the same as those in the variety generated by the two element Boolean algebra. The particular properties that we study here are the finite axiomatizability of their theories, the quantifier complexity of axioms for their theories, and various kinds of model completeness for these theories. It turns out that the proofs of these results can be extracted from the corresponding results for Boolean algebras if one takes the proper viewpoint.Presented by S. Burris.The author acknowledges the useful comments of the editor which led to an improved version of Proposition 1.17 as well as the formulation of some questions at the end of this paper.     

The subdirectly irreducible algebras in the variety generated by graph algebras

We show that every non-trivial subdirectly irreducible algebra in the variety generated by graph algebras is either a two-element left zero semigroup or a graph algebra itself. We characterize all the subdirectly irreducible algebras in this variety. From this we derive an example of a groupoid (graph algebra) that generates a variety with NP-complete membership problem. This is an improvement over the result of Z. Székely who constructed an algebra with similar properties in the signature of two binary operations. The second author was supported by OTKA grants no. T043671, NK67867, K67870 and by NKTH (National Office for Research and Technology, Hungary).     

An effective representation for finitely generated free interior algebras

We give a representation of finitely generated free interior algebras and of the Heyting algebras of their open elements. By means of the representation we give a new and strengthened version of some earlier theorems of McKinsey-Tarski and Blok, etc. regarding such algebras.Presented by S. Burris.