# The equational theory of union-free algebras of relations

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## Abstract

We solve a problem of Jónsson [12] by showing that the class ℛ of (isomorphs of) algebras of binary relations, under the operations of relative product, conversion, and intersection, and with the identity element as a distinguished constant, is not axiomatizable by a set of equations. We also show that the set of equations valid in ℛ is decidable, and in fact the set of equations true in the class of all positive algebras of relations is decidable.

## Keywords

Binary Relation Identity Element Relative Product Equational Theory Distinguished Constant
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## References

- [1]Ackermann, W.,
*Solvable cases of the decision problem*, Studies in logic and the foundations of mathematics, North-Holland Publishing Co., Amsterdam, 1954, viii+114 pp.Google Scholar - [2]Andréka, H.,
*On union-relation composition reducts of relation algebras*, Abstracts of Amer. Math. Soc.*10*,*2*(1989), 174.Google Scholar - [3]Andréka, H.,
*Representation of distributive-lattice-ordered semigroups with binary relations*, Algebra Universalis*28*(1991), 12–25.Google Scholar - [4]Andréka, H. andNémeti, I.,
*Positive reducts of representable relation algebras do not form a variety*, Preprint (1990).Google Scholar - [5]Börner, F. andPöschel, R.,
*Clones of operations on binary relations*, Contributions to general algebra*7*(1991), 51–70.Google Scholar - [6]Bredikhin, D. A.,
*On relation algebras with general superpositions*, in: Algebraic Logic, Coll. Math. Soc. J. Bolyai Vol. 54, North-Holland, 1991, pp. 111–124.Google Scholar - [7]Bredikhin, D. A.,
*The variety generated by ordered involuted semigroups of binary relations*, Proc. Suslin Conf., Saratov, 1991, p. 27.Google Scholar - [8]Bredikhin, D. A.,
*The equational theory of relation algebras with positive operations*(In Russian.) Izv. Vyash, Uchebn. Zaved., Math., No 3, 1993, pp. 23–30.Google Scholar - [9]Bredikhin, D. A. andSchein, B. M.,
*Representations of ordered semigroups and lattices by binary relations*, Colloq. Math.*39*(1978), 1–12.Google Scholar - [10]Comer, S. D.,
*A remark on representable positive cylindric algebras*, Algebra Universalis*28*(1991), 150–151.Google Scholar - [11]Haiman, M.,
*Arguesian lattices which are not linear*, Bull. Amer. Math. Soc.*16*(1987), 121–123.Google Scholar - [12]Jónsson, B.,
*Representation of modular lattices and relation algebras*, Trans. Amer. Math. Soc.*92*(1959), 449–464.Google Scholar - [13]Kozen, D.,
*On induction vs*^{*}-*continuity*, in:*Logics of Programs*, Lecture Notes in Computer Science 131, Springer Verlag, Berlin, 1982, pp. 167–176.Google Scholar - [14]Németi, I.,
*Algebraizations of quantifier logics, An introductory overview, Version 10.2*, Preprint, Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 1992, Abstracted in: Studia Logica, vol L, No 3/4, 1991.Google Scholar - [15]Schein, B. M.,
*Relation algebras and function semigroups*, Semigroup Forum*1*(1970), 1–62.Google Scholar - [16]Schein, B. M.,
*Representation of involuted semigroups by binary relations*, Fund. Math.*82*(1974), 121–141.Google Scholar - [17]Tarski, A. andGivant, S.,
*A formalization of set theory without variables*, Colloquium Publications 41, American Mathematical Society, Providence, R.I., 1987, xxii +318 pp.Google Scholar

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© Birkhäuser Verlag 1995