scholarly journals Congruences and Green's relations on regular semigroups

1972 ◽  
Vol 13 (2) ◽  
pp. 167-175 ◽  
Author(s):  
T. E. Hall
Keyword(s):  
Generalized Inverse ◽  
Inverse Semigroups ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
Green's Relations ◽  
Generalized Inverse Semigroups

It is sometimes possible to reconstruct semigroups from some of their homomorphic images. Some recent examples have been the construction of bisimple inverse semigroups from fundamental bisimple inverse semigroups [9], and the construction of generalized inverse semigroups from inverse semigroups [12].

scholarly journals Congruences on bicyclic extensions of a linearly ordered group

2020 ◽  
Vol 15 (2) ◽  
pp. 61-80
Author(s):  
Oleg Gutik ◽  
Dušan Pagon ◽  
Kateryna Pavlyk
Keyword(s):  
Positive Cone ◽  
Inverse Semigroups ◽  
Green’S Relations ◽  
Ordered Group ◽  
Green's Relations ◽  
Linearly Ordered Group

In the paper we study inverse semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G) which are generated by partial monotone injective translations of a positive cone of a linearly ordered group G. We describe Green’s relations on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G), their bands and show that they are simple, and moreover, the semigroups B(G) and B^+(G) are bisimple. We show that for a commutative linearly ordered group G all non-trivial congruences on the semigroup B(G) (and B^+(G)) are group congruences if and only if the group G is archimedean. Also we describe the structure of group congruences on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G).

REPRESENTATIONS OF LOCALLY INVERSE *-SEMIGROUPS

1996 ◽  
Vol 06 (05) ◽  
pp. 541-551
Author(s):  
TERUO IMAOKA ◽  
ISAMU INATA ◽  
HIROAKI YOKOYAMA
Keyword(s):  
Inverse Semigroup ◽  
Wreath Product ◽  
Representation Theorem ◽  
Generalized Inverse ◽  
Inverse Semigroups ◽  
Locally Inverse Semigroup ◽  
The One ◽  
Locally Inverse Semigroups ◽  
Generalized Inverse Semigroups

The first author obtained a generalization of Preston-Vagner Representation Theorem for generalized inverse *-semigroups. In this paper, we shall generalize their results for locally inverse *-semigroups. Firstly, by introducing a concept of a π-set (which is slightly different from the one in [7]), we shall construct the π-symmetric locally inverse *-semigroup on a π-set, and show that any locally inverse *-semigroup can be embedded up to *-isomorphism in the π-symmetric locally inverse semigroup on a π-set. Moreover, we shall obtain that the wreath product of locally inverse *-semigroups is also a locally inverse *-semigroup.

Normal bands of commutative cancellative semigroups revisited

2014 ◽  
Vol 24 (05) ◽  
pp. 531-551
Author(s):  
Mario Petrich
Keyword(s):  
Free Semigroup ◽  
Abelian Groups ◽  
Normal Band ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Special Cases ◽  
Green's Relations ◽  
Completely Regular Semigroups

A semigroup S is of the type in the class of the title if S has a congruence ρ such that S/ρ is a normal band (i.e. satisfies the identities x2 = x and axya = ayxa) and all ρ-classes are commutative cancellative semigroups. We consider semigroups S with such a congruence first for completely regular semigroups, then characterize the general case in several ways, including some special cases. When S is an order in a normal band of abelian groups Q, we study the restrictions of Green's relations on Q to S. The paper concludes with the discussion of a free semigroup in the title on two generators.

Naturally ordered regular semigroups with a greatest idempotent

1981 ◽  
Vol 91 (1-2) ◽  
pp. 107-122 ◽  
Author(s):  
T. S. Blyth ◽  
R. McFadden
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Natural Order ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
Green's Relations ◽  
Structure Theorems ◽  
Necessary And Sufficient

SynopsisWe consider ordered regular semigroups in which the order extends the natural order on the idempotents, and which are graced with the presence of a greatest idempotent. This implies that every element has a greatest inverse. An investigation into the properties ofthese special elements allows a description of Green's relations on the subsemigroup generated by the idempotents. This in turn leads to a complete description of the structure of idempotent-generated naturally ordered regular semigroups having a greatest idempotent. The smallest such semigroup that is not orthodox is also described. These results lead us to obtain structure theorems in the general case with the added condition that Green's relations be regular. Finally, necessary and sufficient conditions for such a semigroup to be a Dubreil-Jacotin semigroup are found.

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