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Rectangular groupoids and related structures.
Discrete Math, 2013 Boykett T.
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Congruences on *-Regular Semigroups
Periodica Mathematica Hungarica, 2002By a *-regular semigroup \(S\) the authors mean a semigroup with involution * admitting a Moore-Penrose inverse; that is, for each \(a\in S\) there exists a (necessarily unique) solution \(x\) to the equations \(axa=a\), \(xax=x\), \((ax)^*=ax\), \((xa)^*=xa\) which is denoted by \(x=a^+\).
Crvenković, Siniša, Dolinka, Igor
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Regular Orthocryptou Semigroups
Semigroup Forum, 2004The semigroups in this paper are defined using two kinds of generalized Green's relations defined elsewhere. A semigroup \(S\) is superabundant if each \(H^*\)-class contains an idempotent and \(S\) is semisuperabundant if both each \(\widetilde L\)- and \(\widetilde R\)-class contains at least one idempotent. A semigroup is a \(u\)-semigroup if it has
Wang, Zhengpan, Zhang, Ronghua, Xie, Mu
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Acta Mathematica Sinica, English Series, 2004
A semigroup \(S\) is called a weak regular *-semigroup if it has a unary operation * satisfying \[ xx^*x=x,\;(x^*)^*=x,\text{ and }(xx^*yy^*)^*=yy^*xx^*\text{ for all }x,y\text{ in }S. \] In this paper a type of partial algebra called a projective partial groupoid is defined.
Li, Yonghua, Kan, Haibin, Yu, Bingjun
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A semigroup \(S\) is called a weak regular *-semigroup if it has a unary operation * satisfying \[ xx^*x=x,\;(x^*)^*=x,\text{ and }(xx^*yy^*)^*=yy^*xx^*\text{ for all }x,y\text{ in }S. \] In this paper a type of partial algebra called a projective partial groupoid is defined.
Li, Yonghua, Kan, Haibin, Yu, Bingjun
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Variants of Regular Semigroups
Semigroup Forum, 2001Let \(S\) be a semigroup and \(a\in S\); the semigroup with underlying set \(S\) and multiplication \(\circ\) defined by \(x\circ y=xay\) is a variant of \(S\), denoted \((S,a)\). An element of a regular semigroup is regularity preserving if \((S,a)\) is regular.
Khan, T. A., Lawson, M. V.
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Applied Categorical Structures, 2003
Let \(\mathbf C\) be a category with vertex set \(V\) and arrow set \(A\). For \(a\in A\), \(a\sigma\in V\) is the source of \(a\) and \(a\tau\in V\) is the target of \(a\). A flow of \(\mathbf C\) is a mapping \(\varphi\colon V\to A\) such that \((x\varphi)\sigma=x\) for all \(x\in V\).
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Let \(\mathbf C\) be a category with vertex set \(V\) and arrow set \(A\). For \(a\in A\), \(a\sigma\in V\) is the source of \(a\) and \(a\tau\in V\) is the target of \(a\). A flow of \(\mathbf C\) is a mapping \(\varphi\colon V\to A\) such that \((x\varphi)\sigma=x\) for all \(x\in V\).
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Semigroup Forum, 1999
A regular \(*\)-semigroup is a semigroup \(S\) endowed with a supplementary operation \(*\) satisfying: (1) \(xx^*=x\), for every \(x\in S\); (2) \((x^*)^*=x\), for every \(x\in S\); (3) \((xy)^*=y^*x^*\), for every \(x,y\) in \(S\). It has been proved by \textit{M.
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A regular \(*\)-semigroup is a semigroup \(S\) endowed with a supplementary operation \(*\) satisfying: (1) \(xx^*=x\), for every \(x\in S\); (2) \((x^*)^*=x\), for every \(x\in S\); (3) \((xy)^*=y^*x^*\), for every \(x,y\) in \(S\). It has been proved by \textit{M.
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ORTHODOX TRANSVERSALS OF REGULAR SEMIGROUPS
International Journal of Algebra and Computation, 2001Orthodox transversals were introduced by the first author as a generalization of inverse transversals [Comm. Algebra 27(9) (1999), pp. 4275–4288]. One of our aims in this note is to consider the general case of orthodox transversals. The main results are on the sets I and Λ, two components of regular semigroups with orthodox transversals.
J. F. Chen, Y. Q. Cuo
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T-Classification of Regular Semigroups
Semigroup Forum, 2002The author proposes a scheme to classify regular semigroups and takes a first step in that direction. On the congruence lattice of any regular semigroup \(S\) are defined two relations, \(K\) and \(T\), the latter a congruence, the former only meet-preserving, in general.
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A class of regular semigroups with regular *- transversals
Semigroup Forum, 2002A regular semigroup \(S\) is called a regular \(*\)-semigroup if there is a unary operation \(*\) which satisfies the following three conditions: (i) \(xx^*x=x\), (ii) \((x^*)^*=x\), and (iii) \((xy)^*=y^*x^*\), for any \(x,y\in S\) [\textit{T. E. Nordahl, H. E. Scheiblich}, Semigroup Forum 16, 369-377 (1978; Zbl 0408.20043)]. If there a subsemigroup \(
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