free inverse monoid
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2019 ◽  
Vol 29 (08) ◽  
pp. 1467-1498 ◽  
Author(s):  
Ganna Kudryavtseva

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].


2013 ◽  
Vol 23 (08) ◽  
pp. 1789-1803 ◽  
Author(s):  
EMANUELE RODARO ◽  
PEDRO V. SILVA

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.


2009 ◽  
Vol 19 (04) ◽  
pp. 527-554 ◽  
Author(s):  
JOHN FOUNTAIN ◽  
GRACINDA M. S. GOMES ◽  
VICTORIA GOULD

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.


2009 ◽  
Vol 86 (1) ◽  
pp. 27-31 ◽  
Author(s):  
M. J. CRABB

AbstractLet S be an ideal of the free inverse monoid on a set X, and let ℬ denote the Banach algebra l1(S). It is shown that the following statements are equivalent: ℬ is *-primitive; ℬ is prime; X is infinite. A similar result holds if ℬ is replaced by ℂ[S], the complex semigroup algebra of S.


2007 ◽  
Vol 17 (04) ◽  
pp. 761-795 ◽  
Author(s):  
TIMOTHY DEIS ◽  
JOHN MEAKIN ◽  
G. SÉNIZERGUES

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.


2004 ◽  
Vol 14 (05n06) ◽  
pp. 785-799 ◽  
Author(s):  
PEDRO V. SILVA

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.


1998 ◽  
Vol 08 (05) ◽  
pp. 533-551 ◽  
Author(s):  
O. Neto ◽  
H. Sezinando

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.


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

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

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