On semidirect product of semigroups
Abstract
Let X be a subset of a semigroup S. We denote by E(X) the set of idempotent elements Of X.An element a of a semigroup S is called E-inverse if there exists
such that
. We note that the definition is not one-sided. Indeed, an a element of a semigroup S is E-inversive if there exists
such that
(see [7], [l] p. 98). A semigroup S is called E-inversive if all its elements are E-inversive. This class of semigroups is extensive. All semigroups with a zero and all eventually regular semigroups [2] are E-inversive semigroups.Recently E-inversive semigroups reappeared in a paper by Hall and Munn [3] and in a paper by Mitsch [5]. The special case of E-inversive semigroups with pairwise commuting idempotents, called E-dense, was considered by Margolis and Pin [4]. Let S and T be semigroups, and let
be a homomorphism of S into the endomorphism semigroup of T. If
and
, denote
. Thus, if
and
then
. The semidirect product of S and T ,in that order, with strutture map (Y, consists of the set S x T equipped with the product
. This product will be denoted by S _{𝛼}T. In this note we determine which semidirect products of semigroups are E-inversive semigroups and E-dense semigroups, respectively. It turns out that the case in which S induces only automorphism on T allows a particularly simple description. In [6], Preston has answered the analous question for regular semigroups and for inverse semigroups. For the terminology and for the definitions of the algebraic theory of semigroups, we refer to [1].
![x ∈ E(S)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/1f5753772173ea1a232d56b98a818dfa.png)
![ax ∈ E(S)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dcc0c97a00c6d2aea858d3ec45fe1bce.png)
![y ∈ S](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f492d16d15bffee068c5e548ba5e6ea6.png)
![ay, ya ∈ E(S)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/df76e87034e3df35f947621f781a14e7.png)
![𝛼 : S → End(T)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/21552708b7412ffd46b4e6130c5d35af.png)
![s ∈ S](http://siba-ese.unile.it/plugins/generic/latexRender/cache/21e69fc50b1950987f2de85d876f942e.png)
![t ∈ T](http://siba-ese.unile.it/plugins/generic/latexRender/cache/bed7090f3f1924d5941fd1176d897778.png)
![t(sa) by t<sup>s</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/918c2a2a1d9356253acba4f61128ae48.png)
![s,s' ∈ S](http://siba-ese.unile.it/plugins/generic/latexRender/cache/d27e7d60289113d0031cc765e420519f.png)
![t ∈ T](http://siba-ese.unile.it/plugins/generic/latexRender/cache/bed7090f3f1924d5941fd1176d897778.png)
![{t<sup>s</sup>}<sup>s</sup> = t<sup>ss'</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/b5db8e9834635b589b405935e3b66750.png)
![(s,t)(s',t') = (ss', {t<sup>s'</sup>}{t'})](http://siba-ese.unile.it/plugins/generic/latexRender/cache/c1f7fb24930f7b4104a7a82dffee2b9c.png)
DOI Code:
10.1285/i15900932v9n2p189
Full Text: PDF