Semigroup algebras of finite ample semigroups

2012 ◽  
Vol 142 (2) ◽  
pp. 371-389 ◽  
Author(s):  
Xiaojiang Guo ◽  
Lin Chen
Keyword(s):  
Triangular Matrix ◽  
Inverse Semigroups ◽  
Semigroup Algebras ◽  
Matrix Representations ◽  
Ample Semigroup ◽  
Adequate Semigroup

An adequate semigroup S is called ample if ea = a(ea)* and ae = (ae)†a for all a ∈ S and e ∈ E(S). Inverse semigroups are exactly those ample semigroups that are regular. After obtaining some characterizations of finite ample semigroups, it is proved that semigroup algebras of finite ample semigroups have generalized triangular matrix representations. As applications, the structure of the radicals of semigroup algebras of finite ample semigroups is obtained. In particular, it is determined when semigroup algebras of finite ample semigroup are semiprimitive.

MATRIX REPRESENTATIONS OF AMPLE SEMIGROUPS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350108 ◽  
Author(s):  
XIAOJIANG GUO ◽  
K. P. SHUM
Keyword(s):  
Inverse Semigroups ◽  
Matrix Representations ◽  
Ample Semigroup ◽  
Adequate Semigroup ◽  
Brandt Semigroups

An adequate semigroup S is said to be ample if for any e2 = e, a ∈ S, ae = (ae)†a and ea = a(ea)*. It is well known that inverse semigroups are ample semigroups. The purpose of this paper is to study matrix representations of an ample semigroup. Some properties of ample semigroups are obtained. It is proved that any indecomposable good matrix representations of an ample semigroup can be constructed by using those of weak Brandt semigroups.

Locally ample semigroup algebras

2014 ◽  
Vol 07 (04) ◽  
pp. 1450067 ◽  
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Ample Semigroup

An idempotent-connected abundant semigroup S is a locally ample semigroup if for any idempotent e of S, the local submonoid eSe of S is an ample subsemigroup of S. Clearly, an ample semigroup is a locally ample semigroup. In this paper, it is proved that the semigroup algebra of a finite locally ample semigroup is isomorphic to the semigroup algebra of an associate primitive abundant semigroup. As an application of this result, we characterize Jacobson radicals of finite locally ample semigroup algebras.

scholarly journals FREE ADEQUATE SEMIGROUPS

2011 ◽  
Vol 91 (3) ◽  
pp. 365-390 ◽  
Author(s):  
MARK KAMBITES
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Explicit Description ◽  
Finitely Generated ◽  
Free Objects ◽  
Ample Semigroup ◽  
Free Inverse Semigroup ◽  
Adequate Semigroup

AbstractWe give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial ‘folding’ operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are 𝒥-trivial and never finitely generated as semigroups, and that those which are finitely generated as (2,1,1)-algebras have decidable word problem.

Amenability of inverse semigroups and their semigroup algebras

1978 ◽  
Vol 80 (3-4) ◽  
pp. 309-321 ◽  
Author(s):  
J. Duncan ◽  
I. Namioka
Keyword(s):  
Group Algebra ◽  
Inverse Semigroups ◽  
Semigroup Algebras ◽  
Complete Answer ◽  
The Relationship

SynopsisIf G is a group, then G is amenable as a semigroup if and only if l1(G), the group algebra, is amenable as an algebra. In this note, we investigate the relationship between these two notions of amenability for inverse semigroups S. A complete answer can be given in the case where the set Es of idempotent elements of S is finite. Some partial results are obtained for inverse semigroups S with infinite Es.

scholarly journals PARTIAL ACTIONS OF INVERSE AND WEAKLY LEFT E-AMPLE SEMIGROUPS

2009 ◽  
Vol 86 (3) ◽  
pp. 355-377 ◽  
Author(s):  
VICTORIA GOULD ◽  
CHRISTOPHER HOLLINGS
Keyword(s):  
Unary Operation ◽  
Inverse Semigroups ◽  
Partial Transformation ◽  
Transformation Semigroups ◽  
Partial Action ◽  
Ample Semigroup

AbstractWe introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α↦α+, where α+ is the identity map on the domain of α. We investigate the construction of ‘actions’ from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.

scholarly journals Triangular Matrix Representations

2000 ◽  
Vol 230 (2) ◽  
pp. 558-595 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Henry E. Heatherly ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Triangular Matrix ◽  
Matrix Representations

Trace functions on the algebras of certain E-unitary inverse semigroups

1995 ◽  
Vol 125 (5) ◽  
pp. 1077-1084 ◽  
Author(s):  
M. J. Crabb ◽  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebra ◽  
Inverse Semigroups ◽  
Trace Function ◽  
Complex Field ◽  
Semigroup Algebras ◽  
Complex Conjugation ◽  
Arbitrary Rank

A construction is given for a trace function on the semigroup algebra of a certain type of E-unitary inverse semigroup over any subfield of the complex field that is closed under complex conjugation. In particular, the method applies to the semigroup algebras of free inverse semigroups of arbitrary rank.

Matrix representations of right ample semigroups

2015 ◽  
Vol 08 (03) ◽  
pp. 1550042 ◽  
Author(s):  
Junying Guo ◽  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Matrix Representation ◽  
Matrix Representations ◽  
Ample Semigroup ◽  
The Matrix

The properties of right ample semigroups have been extensively considered and studied by many authors. In this paper, we concentrate on the matrix representations of right ample semigroups. The (left; right) uniform matrix representation is initially defined. After some properties of left uniform matrix representations of a right ample semigroup are given, we prove that any irreducible left uniform representations of a right ample semigroup can be obtained by using an irreducible left uniform representation of some primitive right ample semigroup. In particular, a construction theorem of prime left uniform representation of right ample semigroups is established.

On the l1-algebra of certain monoids

1998 ◽  
Vol 128 (5) ◽  
pp. 1023-1031
Author(s):  
M. J. Crabb ◽  
W. D. Munn
Keyword(s):  
Free Monoid ◽  
Explicit Construction ◽  
Semigroup Algebras ◽  
Matrix Representations ◽  
Complex Semigroup

The monoids considered are the free monoid Mx and the free monoid-with-involution MIx on a nonempty set X. In each case, relative to a simply-defined involution, an explicit construction is given for a separating family of continuous star matrix representations of the l1-algebra of the monoid and it is shown that this algebra admits a faithful trace. The results are based on earlier work by M. J. Crabb et al. concerning the complex semigroup algebras of Mx and MIx.

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