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Structure of free strong doppelsemigroups. (English) Zbl 1436.08009
An algebra \({\mathcal D} = (D, \dashv , \vdash )\) with two binary associative operations \( \dashv \), \( \vdash \) satisfying axioms \(x \dashv (y \vdash z) = (x \dashv y) \vdash z\), \(x \vdash (y \dashv z) = (x \vdash y) \dashv z\) is called doppelsemigroup and strong doppelsemigroup if \(x \dashv (y \vdash z) = (x \vdash y) \dashv z\) also holds. An element 0 of a doppelsemigroup \({\mathcal D}\) is zero if \(x * 0 = 0 * x = 0\) for all \(x \in D\), \( * \in \{ \dashv , \vdash \} \). A doppelsemigroup \({\mathcal D}\) with zero is called \(n\)-dinilpotent if \((D, \dashv )\), \((D, \vdash )\) are \(n\)-nilpotent semigroups; \({\mathcal D}\) is called nilpotent if \({x_1}{ * _1}{x_2}{ * _2}\cdots{ * _n}{x_{n + 1}} = 0\) for some \(n \in \mathbb{N}\) and any \({x_i} \in D\), \({ * _j} \in \{ \dashv , \vdash \} \); \({\mathcal D}\) is called commutative if both semigroups \((D, \dashv )\), \((D, \vdash )\) are commutative. Here, constructions for free strong doppelsemigroups, free \(n\)-dinilpotent strong doppelsemigroups, free commutative strong doppelsemigroups and free \(n\)-nilpotent strong doppelsemigroup are presented, the least \(n\)-dinilpotent congruence, the least commutative congruence and the least \(n\)-nilpotent congruence on a free strong doppelsemigroup described and it is shown, that the automorphism group of every constructed free algebra is isomorphic to the symmetric group on the base set.

MSC:
08B20 Free algebras
20M10 General structure theory for semigroups
20M50 Connections of semigroups with homological algebra and category theory
17A30 Nonassociative algebras satisfying other identities
20M75 Generalizations of semigroups
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