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Derived varieties and derived equational theories. (English) Zbl 0940.08003

Summary: This paper describes a derivation process for varieties and equational theories using the theory of hypersubstitutions and \(M\)-hyperidentities. A hypersubstitution \(\sigma\) of type \(\tau\) is a map which takes each \(n\)-ary operation symbol of the type to an \(n\)-ary term of this type. If \(\underline A=(A;(f_i^{\underline A})_{i\in I})\) is an algebra of type \(\tau\) then the algebra \(\sigma (\underline A)=(A;(\sigma (f_i)^{\underline A})_{i\in I})\) is called a derived algebra of \(\underline A\). If \(V\) is a class of algebras of type \(\tau\) then one can consider the variety \(v_\sigma(V)\) generated by the class of all derived algebras from \(V\). In the first two sections the necessary definitions are given. In Section 3 the properties of derived varieties and derived equational theories are described. On the set of all derived varieties of a given variety, a quasiorder is developed which gives a derivation diagram. In the final section the derivation diagram for the largest solid variety of medial semigroups is worked out.

MSC:

08B15 Lattices of varieties
20M07 Varieties and pseudovarieties of semigroups
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[1] DOI: 10.1007/BF02573671 · Zbl 0797.20045 · doi:10.1007/BF02573671
[2] Graczyriska E., Algebra Universalis 2 pp 7– (1990)
[3] Plonka J., Palacky University Olomouc pp 106– (1994)
[4] DOI: 10.1007/BF02188010 · Zbl 0491.08009 · doi:10.1007/BF02188010
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