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A new approach in the theory of orthodox semigroups. (English) Zbl 0705.20052

A bivariety of orthodox semigroups is a class of orthodox semigroups which is closed for taking direct products, regular subsemigroups and homomorphic images. Let X be a countably infinite set and \(F^*(X)\) the free semigroup with involution * on X. A biidentity is a pair \(f\hat=g\) with \(f,g\in F^*(X)\). A multiplicative homomorphism \(\theta\) : \(F^*(X)\to S\) into an orthodox semigroup S is said to be matched if \(x\theta\) and \(x^*\theta\) are mutually inverses for every \(x\in X\). The biidentity \(f\hat=g\) is said to be satisfied in S if \(f\theta =g\theta\) for every matched homomorphism \(\theta\) : \(F^*(X)\to S\). It is shown that a class \({\mathcal V}\) of orthodox semigroups is a bivariety if and only if there exists a set \(\Sigma\) of biidentities such that \({\mathcal V}\) consists precisely of the orthodox semigroups which satisfy all the biidentities in \(\Sigma\).
For a bivariety \({\mathcal V}\) let \(\rho\) (\({\mathcal V},X)\) be the set of all the biidentities which are satisfied in \({\mathcal V}\). Then \(\rho\) (\({\mathcal V},X)\) is a multiplicative congruence on \(F^*(X)\), and the quotient \(F^*(X)/\rho ({\mathcal V},X)=F{\mathcal V}(X)\in {\mathcal V}\) is called bifree for it satisfies the following universal property: if \(T\in {\mathcal V}\) and \(\theta\) : \(X\cup X^*\to T\) is a mapping such that \(x\theta\) and \(x^*\theta\) are mutually inverse in T for every \(x\in X\), then there exists a unique homomorphism \(\phi\) : F\({\mathcal V}(X)\to T\) such that \(\iota \phi =\theta\), where \(\iota\) denotes the obvious mapping \(X\cup X^*\to F{\mathcal V}(X).\)
A class of orthogroups is a bivariety if and only if it is a variety of orthogroups. For any such bivariety \({\mathcal V}\), the free object on X in \({\mathcal V}\) is a subsemigroup of the corresponding bifree object. A solution of the word problem for these bifree objects is obtained, modulo the solution of the word problem for some relatively free groups.
Several sections of the paper are devoted to the bivarieties \(<{\mathcal A}\circ {\mathcal K}>\) generated by the Mal’cev product \({\mathcal A}\circ {\mathcal K}\) within the class of all orthodox semigroups of a variety of bands \({\mathcal A}\) and a variety of groups \({\mathcal K}\). The bifree objects in \(<{\mathcal A}\circ {\mathcal K}>\) are E-unitary and the word problem is reduced to the word problem for the free groups in \({\mathcal K}\). A model for the bifree object \(F<{\mathcal A}\circ {\mathcal K}>(X)\) is constructed: it is shown that \(F<{\mathcal A}\circ {\mathcal K}>(X)\) is a subsemigroup of a semidirect product of a bifree band in \({\mathcal A}\) by F\({\mathcal K}(X).\)
A bifree object in a self-dual variety of orthogroups is exactly a free object in a variety of *-orthogroups. This entails that there exists a one-to-one correspondence between the varieties of *-orthogroups and the self-dual varieties of orthogroups. Also, if \({\mathcal A}^*\circ {\mathcal K}\) is the Mal’cev product within the class of *-orthodox semigroups of a variety \({\mathcal A}^*\) of *-bands and a variety \({\mathcal K}\) of groups, then the free object in the variety \(<{\mathcal A}^*\circ {\mathcal K}>\) is the bifree object in the bivariety \(<{\mathcal A}\circ {\mathcal K}>\), \({\mathcal A}\) being the self-dual band variety corresponding to \({\mathcal A}^*\).
Reviewer: F.Pastijn

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems
20M19 Orthodox semigroups
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