The Centre of the Algebra of a Free Inverse Monoid

1997 ◽  
Vol 55 (2) ◽  
pp. 215-220 ◽  
Author(s):  
M.J. Crabb ◽  
W.D. Munn
Keyword(s):  
Inverse Monoid ◽  
Free Inverse Monoid

FREE INVERSE MONOIDS AND GRAPH IMMERSIONS

1993 ◽  
Vol 03 (01) ◽  
pp. 79-99 ◽  
Author(s):  
STUART W. MARGOLIS ◽  
JOHN C. MEAKIN
Keyword(s):  
Formal Language ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Rational Subset ◽  
The Relationship ◽  
Subgroup Theorem ◽  
Free Inverse Monoid ◽  
Natural Way

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

On The Schützenberger Product of Three Copies of a Free Group

1998 ◽  
Vol 08 (05) ◽  
pp. 533-551 ◽  
Author(s):  
O. Neto ◽  
H. Sezinando
Keyword(s):  
Free Group ◽  
Free Monoid ◽  
Inverse Monoid ◽  
Natural Structure ◽  
Free Inverse Monoid

We show that the Schützenberger product "cut down to generators" of three copies of a free group is a relatively free monoid with involution. We show that its set of idempotents has a natural structure of a semilattice. This semilattice is naturally isomorphic to the semilattice of idempotents of a free inverse monoid.

THE FREE AMPLE MONOID

2009 ◽  
Vol 19 (04) ◽  
pp. 527-554 ◽  
Author(s):  
JOHN FOUNTAIN ◽  
GRACINDA M. S. GOMES ◽  
VICTORIA GOULD
Keyword(s):  
Semidirect Product ◽  
Free Monoid ◽  
Inverse Monoid ◽  
Free Inverse Monoid

We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y*T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X* acts doubly on the semilattice [Formula: see text] of idempotents of FIM(X) and that FAM(X) is embedded in [Formula: see text]. Finally we show that every weakly E-ample monoid has a proper ample cover.

Two-sided expansions of monoids

2019 ◽  
Vol 29 (08) ◽  
pp. 1467-1498 ◽  
Author(s):  
Ganna Kudryavtseva
Keyword(s):  
Inverse Monoid ◽  
Partial Action ◽  
Expansion Formula ◽  
Inverse Monoids ◽  
Universal Formula ◽  
Special Case ◽  
Free Inverse Monoid

We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget–Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of relatively free inverse monoids. For a monoid [Formula: see text] and a class of partial actions of [Formula: see text], determined by a set, [Formula: see text], of identities, we define [Formula: see text] to be the universal [Formula: see text]-generated two-sided restriction monoid with respect to partial actions of [Formula: see text] determined by [Formula: see text]. This is an [Formula: see text]-restriction monoid which (for a certain [Formula: see text]) generalizes the Birget–Rhodes prefix expansion [Formula: see text] of a group [Formula: see text]. Our main result provides a coordinatization of [Formula: see text] via a partial action product of the idempotent semilattice [Formula: see text] of a similarly defined inverse monoid, partially acted upon by [Formula: see text]. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result. We show that some properties of [Formula: see text] agree well with suitable properties of [Formula: see text], such as being cancellative or embeddable into a group. We observe that if [Formula: see text] is an inverse monoid, then [Formula: see text], the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson–Margolis–Steinberg generalized prefix expansion [Formula: see text]. This gives a presentation of [Formula: see text] and leads to a model for [Formula: see text] in terms of the known model for [Formula: see text].

AN APPLICATION OF FIRST-ORDER LOGIC TO THE STUDY OF RECOGNIZABLE LANGUAGES

2004 ◽  
Vol 14 (05n06) ◽  
pp. 785-799 ◽  
Author(s):  
PEDRO V. SILVA
Keyword(s):  
Finite Rank ◽  
Free Monoid ◽  
Order Logic ◽  
First Order Logic ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
First Order ◽  
Free Inverse Monoid

A variation of first-order logic with variables for exponents is developed to solve some problems in the setting of recognizable languages on the free monoid, accommodating operators such as product, bounded shuffle and reversion. Restricting the operators to powers and product, analogous results are obtained for recognizable languages of an arbitrary finitely generated monoid M, in particular for a free inverse monoid of finite rank. As a consequence, it is shown to be decidable whether or not a recognizable subset of M is pure or p-pure.

scholarly journals On the Algebra of a Free Inverse Monoid

1996 ◽  
Vol 184 (1) ◽  
pp. 297-303 ◽  
Author(s):  
M.J. Crabb ◽  
W.D. Munn
Keyword(s):  
Inverse Monoid ◽  
Free Inverse Monoid

scholarly journals ON PERIODIC POINTS OF FREE INVERSE MONOID HOMOMORPHISMS

2013 ◽  
Vol 23 (08) ◽  
pp. 1789-1803 ◽  
Author(s):  
EMANUELE RODARO ◽  
PEDRO V. SILVA
Keyword(s):  
Fixed Point ◽  
Periodic Point ◽  
Periodic Points ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Context Sensitive ◽  
Context Free Language ◽  
Context Free ◽  
Free Language ◽  
Free Inverse Monoid

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.

scholarly journals On the semilattice of idempotents of a free inverse monoid

1993 ◽  
Vol 36 (2) ◽  
pp. 349-360
Author(s):  
Pedro V. Silva
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Monoid ◽  
Unique Factorization ◽  
Principal Ideals ◽  
New Concepts ◽  
Necessary And Sufficient ◽  
Free Inverse Monoid

Some new concepts are introduced, in particular that of a unique factorization semilattice. Necessary and sufficient conditions are given for two principal ideals of the semilattice of idempotents of a free inverse monoid FIM(X) to be isomorphic and some properties of the Munn semigroup of E[FIM(X)] are obtained. Some results on the embedding of semilattices in E[FIM(X)] are also obtained.

scholarly journals EQUATIONS IN FREE INVERSE MONOIDS

2007 ◽  
Vol 17 (04) ◽  
pp. 761-795 ◽  
Author(s):  
TIMOTHY DEIS ◽  
JOHN MEAKIN ◽  
G. SÉNIZERGUES
Keyword(s):  
Free Group ◽  
Free Monoid ◽  
Finite System ◽  
System Of Equations ◽  
Inverse Monoid ◽  
Single Variable ◽  
Systems Of Equations ◽  
Inverse Monoids ◽  
Consistency Problem ◽  
Free Inverse Monoid

It is known that the problem of determining consistency of a finite system of equations in a free group or a free monoid is decidable, but the corresponding problem for systems of equations in a free inverse monoid of rank at least two is undecidable. Any solution to a system of equations in a free inverse monoid induces a solution to the corresponding system of equations in the associated free group in an obvious way, but solutions to systems of equations in free groups do not necessarily lift to solutions in free inverse monoids. In this paper, we show that the problem of determining whether a solution to a finite system of equations in a free group can be extended to a solution of the corresponding system in the associated free inverse monoid is decidable. We are able to use this to solve the consistency problem for certain classes of single-variable equations in free inverse monoids.

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