DUALISABILITY OF FINITE SEMIGROUPS

2003 ◽  
Vol 13 (04) ◽  
pp. 481-497 ◽  
Author(s):  
MARCEL JACKSON
Keyword(s):  
Inverse Semigroups ◽  
Commutative Monoids ◽  
Finite Algebras ◽  
Finite Monoids ◽  
Finite Semigroups ◽  
Entropic Algebra ◽  
Aperiodic Monoids

We describe the inherently non-dualisable finite algebras from some semigroup related classes. The classes for which this problem is solved include the variety of bands, the pseudovariety of aperiodic monoids, commutative monoids, and (assuming a reasonable conjecture in the literature) the varieties of all finite monoids and finite inverse semigroups. The first example of an inherently non-dualisable entropic algebra is also presented.

scholarly journals EMBEDDINGS IN COSET MONOIDS

2008 ◽  
Vol 85 (1) ◽  
pp. 75-80
Author(s):  
JAMES EAST
Keyword(s):  
Semigroup Forum ◽  
Inverse Semigroup ◽  
Simple Proof ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Simple Description ◽  
Group Of Units ◽  
Finite Monoids ◽  
Finite Set ◽  
Symmetric Inverse Semigroup

AbstractA submonoid S of a monoid M is said to be cofull if it contains the group of units of M. We extract from the work of Easdown, East and FitzGerald (2002) a sufficient condition for a monoid to embed as a cofull submonoid of the coset monoid of its group of units, and show further that this condition is necessary. This yields a simple description of the class of finite monoids which embed in the coset monoids of their group of units. We apply our results to give a simple proof of the result of McAlister [D. B. McAlister, ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum20 (1980), 255–267] which states that the symmetric inverse semigroup on a finite set X does not embed in the coset monoid of the symmetric group on X. We also explore examples, which are necessarily infinite, of embeddings whose images are not cofull.

scholarly journals Inverse semigroups and varieties of finite semigroups

1987 ◽  
Vol 110 (2) ◽  
pp. 306-323 ◽  
Author(s):  
S.W Margolis ◽  
J.E Pin
Keyword(s):  
Inverse Semigroups ◽  
Finite Semigroups

scholarly journals Finite semigroups with commuting idempotents

1987 ◽  
Vol 43 (1) ◽  
pp. 81-90 ◽  
Author(s):  
C. J. Ash ◽  
T. E. Hall
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Finite Semigroups

AbstractWe show that every such semigroup is a homomorphic image of a subsemigroup of some finite inverse semigroup. This shows that the pseudovariety generated by the finite inverse semigroups consists of exactly the finite semigroups with commuting idempotents.

scholarly journals Semigroups whose idempotents form a subsemigroup

1990 ◽  
Vol 41 (2) ◽  
pp. 161-184 ◽  
Author(s):  
Jean-Camille Birget ◽  
Stuart Margolis ◽  
John Rhodes
Keyword(s):  
Inverse Semigroup ◽  
Finite Semigroup ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Type Ii ◽  
Finite Semigroups ◽  
Case Type ◽  
Orthodox Semigroups

We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.For these three classes of semigroups, type-II is equal to type-II construct.

Compactness of Systems of Equations in Semigroups

1997 ◽  
Vol 07 (04) ◽  
pp. 457-470 ◽  
Author(s):  
T. Harju ◽  
J. Karhumäki ◽  
W. Plandowski
Keyword(s):  
Necessary Conditions ◽  
Chain Condition ◽  
Inverse Semigroups ◽  
System Of Equations ◽  
Systems Of Equations ◽  
Finitely Generated ◽  
Compactness Property ◽  
Commutative Monoids ◽  
Maximal Condition ◽  
Bicyclic Monoid

We consider systems ui=vi(i∈I) of equations in semigroups over finite sets of variables. A semigroup (or a monoid) S is said to satisfy the compactness property (CP, for short), if each system of equations has an equivalent finite subsystem. We prove that all monoids in a variety [Formula: see text] satisfy CP if and only if the finitely generated monoids in [Formula: see text] satisfy the maximal condition on congruences. Consequently, all commutative monoids (and semigroups) satisfy CP. Also, if a finitely generated semigroup S satisfies CP, then S is necessarily hopfian and satisfies the chain condition on idempotents. It follows that the free inverse semigroups do not satisfy CP. Finally, we give two simple examples (the bicyclic monoid and the Baumslag-Solitar group) which do not satisfy CP, and show that the necessary conditions above are not sufficient.

scholarly journals A PROFINITE APPROACH TO STABLE PAIRS

2010 ◽  
Vol 20 (02) ◽  
pp. 269-285 ◽  
Author(s):  
KARSTEN HENCKELL ◽  
JOHN RHODES ◽  
BENJAMIN STEINBERG
Keyword(s):  
Short Proof ◽  
Finite Monoids ◽  
Aperiodic Monoids ◽  
Stable Pairs

We give a short proof, using profinite techniques, that idempotent pointlikes, stable pairs and triples are decidable for the pseudovariety of aperiodic monoids. Stable pairs are also described for the pseudovariety of all finite monoids.

On Finite Semigroups Embeddable in Inverse Semigroups

2001 ◽  
Vol 62 (2) ◽  
pp. 329-330
Author(s):  
Boris M. Schein
Keyword(s):  
Inverse Semigroups ◽  
Finite Semigroups

On surjunctive monoids

2015 ◽  
Vol 25 (04) ◽  
pp. 567-606 ◽  
Author(s):  
Tullio Ceccherini-Silberstein ◽  
Michel Coornaert
Keyword(s):  
Cellular Automata ◽  
The Other ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Commutative Monoids ◽  
Residually Finite ◽  
Finite Monoids ◽  
Other Hand ◽  
Bicyclic Monoid

A monoid M is called surjunctive if every injective cellular automata with finite alphabet over M is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive.

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