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Relation algebras and Schröder categories. (English) Zbl 0649.03047
This paper begins with the definition of relation algebras $$(A,+,\cdot,\quad -,0,1,;,^{\sqcup},')$$ and a recall of some of their properties. A Schröder category, as defined by Olivier and Serrato, is a category $${\mathfrak C}$$ such that for every i,j$$\in Ob {\mathfrak C}$$, the set $${\mathfrak C}(i,j)$$ is a Boolean algebra and there is an involution $$^{\sqcup}: {\mathfrak C}(i,j)\to {\mathfrak C}(j,i)$$; besides, the composition of morphism and $$^{\sqcup}$$ satisfies certain supplementary conditions. The author proves that if I is any finite set of objects of a Schröder category $${\mathfrak C}$$, then the direct product of the Boolean algebras $${\mathfrak C}(i,j)$$, $$i,j\in I$$, can be made into a relation algebra. Then, a natural concept of tensor product of Boolean algebras is introduced which turns out to be the same as the free product. The next section deals with Boolean modules. Further, given a system of simple relation algebras $${\mathfrak A}_ i$$, $$i\in I$$, the author constructs from them in a canonical way a Schröder category $${\mathfrak C}$$ whose objects are the members of I in such a way that $${\mathfrak C}(i,i)={\mathfrak A}_ i$$, and assuming that I is finite, the semi-product of the algebras $${\mathfrak A}_ i$$ is defined as a certain algebra $${\mathfrak A}$$ of matrices a with entries $$a(i,j)\in {\mathfrak C}(i,j)$$; the algebras $${\mathfrak A}_ i$$ determine $${\mathfrak A}$$ up to isomorphism. The last sections study in some detail the equivalence elements of a relation algebra (i.e., the elements u such that $$u;u\leq u$$ and $$u^{\sqcup}=u)$$ and the relation algebras that are generated by an equivalence element. Every such algebra is finite and representable.
Reviewer: S.Rudeanu

##### MSC:
 03G15 Cylindric and polyadic algebras; relation algebras 18D99 Categorical structures 06E99 Boolean algebras (Boolean rings) 08B25 Products, amalgamated products, and other kinds of limits and colimits
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##### References:
 [1] Brink, C., Boolean modules, J. algebra, 71, 291-313, (1981) · Zbl 0465.03028 [2] Chin, L.H.; Tarski, A., Distributive and modular laws in relation algebras, Univ. of calif. publ. in math., Vol 1, 341-383, (1951), N.S. [3] Jónsson, B.; Tarski, A.; Jónsson, B.; Tarski, A., Boolean algebras with operators, part II, Amer. J. math., Amer. J. math., 74, 127-162, (1952) · Zbl 0045.31601 [4] Jónsson, B., Varieties of relation algebras, Algebra universalis, 15, 273-298, (1982) · Zbl 0545.08009 [5] Olivier, J.-P.; Serrato, D., Catégories de Dedekind. morphisms dans LES catégories de Schröder, C.R. acad. sci. Paris, 290, 939-941, (1980) · Zbl 0438.18003
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