Free Inverse Semigroups

Munn, W. D.

doi:10.1112/plms/s3-29.3.385

Cited by 151 publications

(98 citation statements)

References 3 publications

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“…Then y i+1 (= y i ) is a negatively oriented edge of Y. We have a i = m i t y i , a i+1 = m i+1 by (13), and a i = a i+1 t −1 y i+1 k i+1 and (a i+1 t −1 (14). Then…”

Section: Initial and Terminal Vertexmentioning

confidence: 97%

“…Munn [13] initiated the study of free inverse semigroups in terms of graphs, Jones [8] used graphs to study free products of inverse semigroups, and Stephen [19] introduced Schützenberger graphs to study presentations of inverse semigroups. In this paper, we define a partial order on graphs, as it is introduced in [12], and then we consider its relationship with inverse semigroup presentations.…”

Section: Preliminariesmentioning

confidence: 99%

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A class of inverse monoids acting on ordered forests

Yamamura

2004

Journal of Algebra

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Inverse monoid actions on ordered forests are studied to generalize the Bass-Serre theory to a certain class of inverse monoids. We introduce the concepts of graphs of inverse monoids and their fundamental inverse monoids and discuss their basic properties. Using these concepts, we characterize the inverse monoids acting on ordered forests satisfying some conditions as the groups acting on trees without inversion are characterized as the fundamental groups of graphs of groups. We also investigate the local action of a maximal subgroup of the fundamental inverse monoid on a connected component of the universal cover and obtain a presentation of the maximal subgroup.  2004 Elsevier Inc. All rights reserved.

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“…Then y i+1 (= y i ) is a negatively oriented edge of Y. We have a i = m i t y i , a i+1 = m i+1 by (13), and a i = a i+1 t −1 y i+1 k i+1 and (a i+1 t −1 (14). Then…”

Section: Initial and Terminal Vertexmentioning

confidence: 97%

Section: Preliminariesmentioning

confidence: 99%

A class of inverse monoids acting on ordered forests

Yamamura

2004

Journal of Algebra

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“…The first one is due to Scheiblich [58]. It can be seen as a fairly compact way to represent the second one, more graphical, independently due to Munn [48]. Here, in both cases, elements of the free monoid are called birooted trees.…”

Section: Inverse and Free Inverse Monoidmentioning

confidence: 98%

“…Scheiblich-Munn theorem [58,48] states that FIM( A) is the free inverse monoid generated by A. In other words, for every inverse monoid M, for every mapping ϕ : A → M there is a unique morphism of inverse monoid ψ : FIM( A) → M such that ψ|A equals ϕ.…”

Section: Inverse and Free Inverse Monoidmentioning

confidence: 98%

On labeled birooted tree languages: Algebras, automata and logic

Janin

2015

Information and Computation

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With an aim to developing expressive language theoretical tools applicable to inverse semigroup languages, that is, subsets of inverse semigroups, this paper explores the language theory of finite labeled birooted trees: Munn's birooted trees extended with vertex labeling. To this purpose, we define a notion of finite state birooted tree automata that simply extends finite state word automata semantics. This notion is shown to capture the class of languages that are definable in Monadic Second Order Logic and upward closed with respect to the natural order. Then, we derive from these automata the notion of quasi-recognizable languages, that is, languages recognizable by means of (adequate) premorphisms into finite (adequately) ordered monoids. This notion is shown to capture finite Boolean combinations of languages as above, a class that contains classical regular languages of finite (mono-rooted) trees. This contrasts with the known collapse of classical algebraic tools when applied to inverse semigroups.

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“…(2) By [14] every free inverse semigroup S is "finite J -above", that is, for any x ∈ S there are only finitely many elements y such that SyS ⊆ SxS. Hence no idempotent can be strictly below infinitely many others.…”

Section: Archimedean Propertiesmentioning

confidence: 99%

Inverse semigroups determined by their lattices of convex inverse subsemigroups II

Cheong¹,

Jones

2003

Algebra Universalis

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In Part I of this paper we showed that the study of Co -isomorphisms of inverse semigroups, that is, isomorphisms between the lattices of convex inverse subsemigroups of two such semigroups, can be reduced to consideration of those that induce an isomorphism between the respective semilattices of idempotents. We go on here to prove two somewhat complementary theorems. We show that the inverse semigroups Co -isomorphic to the bicyclic semigroup are precisely the well-known simple, aperiodic, inverse ω-semigroups B d . And we show that for completely semisimple inverse semigroups (those that contain no bicyclic subsemigroup), Co -isomorphisms that induce an isomorphism between the semilattices of idempotents of the respective inverse semigroups are entirely equivalent to L-isomorphisms with the same property. (An L-isomorphism is an isomorphism between the lattices of all inverse subsemigroups.) Combining this result, known results on L-isomorphisms and the main theorem of Part I yields a complete determination of Co -isomorphisms for broad classes of semigroups. For some slightly narrower classes it is known that every Co -isomorphism of necessity induces an isomorphism on the semilattice of idempotents, yielding theorems on their Co -determinability.

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Free Inverse Semigroups

Cited by 151 publications

References 3 publications

A class of inverse monoids acting on ordered forests

A class of inverse monoids acting on ordered forests

On labeled birooted tree languages: Algebras, automata and logic

Inverse semigroups determined by their lattices of convex inverse subsemigroups II

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