The Variety Generated by Equivalence Algebras
نویسندگان
چکیده
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.
منابع مشابه
Dually quasi-De Morgan Stone semi-Heyting algebras I. Regularity
This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras of level 1 satisfies the strongly blended $lor$-De Morgan law introduced in cite{Sa12}. Then, using this result and the results of cite{Sa12}, we prove our main result which gives an explicit description of simple algebras(=subdirectly irreducibles) ...
متن کاملDually quasi-De Morgan Stone semi-Heyting algebras II. Regularity
This paper is the second of a two part series. In this Part, we prove, using the description of simples obtained in Part I, that the variety $mathbf{RDQDStSH_1}$ of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element $mathbf{RDQDStSH_1}$-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras--...
متن کاملar X iv : m at h / 05 09 03 2 v 3 [ m at h . G M ] 1 8 Fe b 20 06 AUTOMORPHIC EQUIVALENCE OF ONE - SORTED ALGEBRAS
One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? There are various interpretations of the sentence " Two algebras have the same algebraic geometry ". One of these is automorphic equivalence of algebras, which is discussed in this paper, and the other interpretation is geometric equivalence of algebras. In this paper we consider...
متن کاملBCK-ALGEBRAS AND HYPER BCK-ALGEBRAS INDUCED BY A DETERMINISTIC FINITE AUTOMATON
In this note first we define a BCK‐algebra on the states of a deterministic finite automaton. Then we show that it is a BCK‐algebra with condition (S) and also it is a positive implicative BCK‐algebra. Then we find some quotient BCK‐algebras of it. After that we introduce a hyper BCK‐algebra on the set of all equivalence classes of an equivalence relation on the states of a deterministic finite...
متن کاملDynamic equivalence relation on the fuzzy measure algebras
The main goal of the present paper is to extend classical results from the measure theory and dynamical systems to the fuzzy subset setting. In this paper, the notion of dynamic equivalence relation is introduced and then it is proved that this relation is an equivalence relation. Also, a new metric on the collection of all equivalence classes is introduced and it is proved that this metric is...
متن کامل