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Simple iterative algebras. (English. Russian original) Zbl 0913.03057

Algebra Logika 37, No. 4, 460-477 (1998); translation in Algebra Logic 37, No. 4, 260-269 (1998).
Since the associative operation of superposition is basic for iterative algebras, it is natural, in particular, to appeal to the semigroup methods in studying them. In the article under review, the author continues this direction of research. Based on the notion of an ideal of an iterative algebra \(A\) (i.e., a subalgebra in \(A\) which is a right ideal in the semigroup \(\langle A; *\rangle\)), the author introduces the notion of a simple iterative algebra of functions of a \(k\)-valued logic, i.e., an algebra that has no proper ideals. The set of ideals of \(A\) is a distributive sublattice of the lattice of subalgebras of \(A\). It is shown that every simple algebra is included in a maximal algebra and is naturally associated with some permutation group on a finite set. Some conditions on a permutation group are given under which the corresponding algebra turns out to be maximal and simple. This allows the author to completely list the maximal simple algebras for the case \(k\leq 4\).

MSC:

03G20 Logical aspects of Łukasiewicz and Post algebras
06D25 Post algebras (lattice-theoretic aspects)
20M12 Ideal theory for semigroups
20M25 Semigroup rings, multiplicative semigroups of rings
68Q45 Formal languages and automata
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