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Abstract  

In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup.

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semigroups , Collect. Math ., 41 ( 1980 ), 189 – 195 . [2] E dwards , P. M. , Eventually regular semigroups , Bull. Austral. Math. Soc ., 28 ( 1983 ), 23 – 38

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References [1] Jones , P. R. , Mal’cev products of varieties of completely regular semigroups , J. Austral. Math

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Abstract  

The class

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of projectively condensed semigroups is a quasivariety of unary semigroups, the class of projective orthomonoids is a subquasivariety of
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. Some well-known classes of generalized completely regular semigroups will be regarded as subquasivarieties of
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. We give the structure semilattice composition and the standard representation of projective orthomonoids, and then obtain the structure theorems of various generalized orthogroups.

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of quasi-adequate semigroups , Comm. Algebra , 40 ( 3 ), 905 - 930 ( 2012 ) [3] Blyth , T. S. and McFadden , R. , Regular semigroups with a multiplicative inverse

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In this paper,P-ordered andQ-ordered semigroups are studied. Some characterizations and properties of such semigroups are obtalned. Also the relationship between maximal (minimum) regular ordered semigroups and unitary regular semigroups is investigated.

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Alimpić, B. P. and Krgović, D. N. , Some congruences on regular semigroups, Lect. Notes Math. , 1320 , Springer-Verlag (1986), 1–10. Krgović D. N

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