# Finitely generated commutative monoids There is a large amount of bibliography related to Semigroup Theory. Apart from Redei's book, almos Some of them have several chapters involving commutative semigroups, but the theory developed there is too general and few of them focus their attention on finitely generated commutative semigroups. Thus when one tries to study this kind of semigroups one has to "swallow'' many concepts that are not suitable and that, in many cases, turn out to be trivial for finitely generated commutative monoids.

In addition, there is a lack of effective methods for studying properties of finitely generated commutative monoids. These were in fact the chief reasons for developing a self-contained book on finitely generated commutative monoids with the theory and algorithms needed for the study of the main classical problems related to this kind of monoid.

This book is not only addressed to people working in Semigroup Theory. The only knowledge required to follow and understand its contents is basic Linear Algebra. Thus any student of second year of Mathematics or Computer Science might find the book easy to understand. This monograph can also be used as a textbook of a course on finitely generated monoids. This material is not only interesting from the semigroup point of view, it has many applications to other fields in algebra, as the theory stems from problems in these fields of research.

Shiny and new! BetterWorldBooks Marketplace. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Svarc-Milnor Lemma to this setting. We study the way in which the abstract structure of a small overlap monoid is reflected in, and may be algorithmically deduced from, a small overlap presentation.

### Some algorithms for finitely generated commutative monoids

We show that every C 2 monoid admits an essentially canonical C 2 presentation; by counting canonical presentations we obtain asymptotic estimates for the number of non-isomorphic monoids admitting a-generator, k-relation presentations of a given length. We demonstrate an algorithm to transform an arbitrary presentation for a C m monoid m at least 2 into a canonical C m presentation, and a solution to the isomorphism problem for C 2 presentations.

We also find a simple combinatorial condition on a C 4 presentation which is necessary and sufficient for the monoid presented to be left cancellative. We apply this to obtain algorithms to decide if a given C 4 monoid is left cancellative, right cancellative or cancellative, and to show that cancellativity properties are asymptotically visible in the sense of generic-case complexity.

Small overlap conditions are simple and natural combinatorial conditions on semigroup and monoid presentations, which serve to limit the complexity of derivation sequences between equivalent words in the generators. They were introduced by J. Remmers, and more recently have been extensively studied by the present author. However, the definition of small overlap conditions hitherto used by the author was slightly more restrictive than that introduced by Remmers; this note eliminates this discrepancy by extending the recent methods and results of the author to apply to Remmers' small overlap monoids in full generality.

We study the algebraic structure of the semigroup of all 2x2 tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green's relations and the idempotents and maximal subgroups of this semigroup. We study groups acting by length-preserving transformations on spaces equipped with asymmetric, partially-defined distance functions.

We introduce a natural notion of quasi-isometry for such spaces and exhibit an extension of the Svarc-Milnor Lemma to this setting. Among the most natural examples of these spaces are finitely generated monoids and semigroups and their Cayley and Schutzenberger graphs; we apply our results to show a number of important properties of monoids are quasi-isometry invariants. Recent research of the author has studied edge-labelled directed trees under a natural multiplication operation. The class of all such trees with a fixed labelling alphabet has an algebraic interpretation, as a free object in the class of adequate semigroups.

In the present paper, we consider a natural subclass of these trees, defined by placing a restriction on edge orientations, and show that the resulting algebraic structure is a free object in the class of left adequate semigroups. Through this correspondence we establish some structural and algorithmic properties of free left adequate semigroups and monoids, and consequently of the category of all left adequate semigroups. This is the final version of a paper, the initial April preprint of which was entitled Free left and right adequate semigroups.

We 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. We study the generic properties of finitely presented monoids and semigroups, and the generic-case complexity of decision problems for them.

We show that for a finite alphabet A of size at least 2 and positive integers k and m, the generic A-generated k-relation monoid and semigroup defined using any of several stratifications satisfies the small overlap condition C m. It follows that the generic finitely presented monoid has a number of interesting semigroup-theoretic properties and, by a recent result of the author, admits a linear time solution to its word problem and a regular language of unique normal forms for its elements.

Moreover, the uniform word problem for finitely presented monoids is generically solvable in polynomial time. We show that any finite monoid or semigroup presentation satisfying the small overlap condition C 4 has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time. We also deduce that C 4 monoids and semigroups are rational in the sense of Sakarovitch , asynchronous automatic, and word hyperbolic in the sense of Duncan and Gilman.

From this it follows that C 4 monoids satisfy analogues of Kleene's theorem, and admit decision algorithms for the rational subset and finitely generated submonoid membership problems. We also prove some automata-theoretic results which may be of independent interest. Our first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product.

## Ideals of finitely generated commutative monoids

Our second main result gives a presentation for the inverse hull of such a graph product. We then specialise to the case of the inverse hulls of graph monoids, obtaining what we call polygraph monoids. We develop a combinatorial approach to the study of semigroups and monoids with finite presentations satisfying small overlap conditions. In contrast to existing geometric methods, our approach facilitates a sequential left-right analysis of words which lends itself to the development of practical, efficient computational algorithms.

In particular, we obtain a highly practical linear time solution to the word problem for monoids and semigroups with finite presentations satisfying the condition C 4 , and a polynomial time solution to the uniform word problem for presentations satisfying the same condition. We study the classes of languages defined by valence automata with rational target sets or equivalently, regular valence grammars with rational target sets , where the valence monoid is drawn from the important class of polycyclic monoids.

We show that for polycyclic monoids of rank 2 or more, such automata accept exactly the context-free languages. For the polycyclic monoid of rank 1 that is, the bicyclic monoid , they accept a class of languages strictly including the partially blind one-counter languages. Key to the proof is a description of the rational subsets of polycyclic and bicyclic monoids, other consequences of which include the decidability of the rational subset membership problem for these monoids, and the closure of the class of rational subsets under intersection and complement.

We show that for any monoid M, the family of languages accepted by M-automata or equivalently, generated by regular valence grammars over M is completely determined by that part of M which lies outside the maximal ideal. Hence, every such family arises as the family of languages accepted by N-automata where N is a simple or 0-simple monoid.

A consequence is that every such family is either the class of regular languages, contains all the blind one-counter languages, or is the family of languages accepted by G-automata for G a non-locally-finite torsion group. We consider a natural extension of the usual definition which permits the automata to utilise more of the structure of each monoid, and also allows us to define S-automata for S an arbitrary semigroup.

In the monoid case, the resulting automata are equivalent to the valence automata with rational target sets which arise in the theory of regulated rewriting systems. We study the case that the register semigroup is completely simple or completely 0-simple, obtaining a complete characterisation of the classes of languages corresponding to such semigroups in terms of their maximal subgroups. In the process, we obtain a number of results about rational subsets of Rees matrix semigroups which may be of independent interest. We study the relationship between the loop problem of a semigroup, and that of a Rees matrix construction with or without zero over the semigroup.

This allows us to characterize exactly those completely zero-simple semigroups for which the loop problem is context-free. We also establish some results concerning loop problems for subsemigroups and Rees quotients. We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the Muller-Schupp theorem.

More generally, if G is a virtually abelian group then every group with word problem recognised by a G-automaton is virtually abelian with growth class bounded above by the growth class of G. We consider also other types of counter automata. We propose a way of associating to each finitely generated monoid or semigroup a formal language, called its loop problem. In the case of a group, the loop problem is essentially the same as the word problem in the sense of combinatorial group theory.

Like the word problem for groups, the loop problem is regular if and only if the monoid is finite. We study also the case in which the loop problem is context-free, showing that a celebrated group-theoretic result of Muller and Schupp extends to describe completely simple semigroups with context-free loop problems.

We consider also right cancellative monoids, establishing connections between the loop problem and the structural theory of these semigroups by showing that the syntactic monoid of the loop problem is the inverse hull of the monoid. A finitely generated group is called a Church-Rosser group growing context-sensitive group if it admits a finitely generated presentation for which the word problem is a Church-Rosser growing context-sensitive language. Although the Church-Rosser languages are incomparable to the context-free languages under set inclusion, they strictly contain the class of deterministic context-free languages.

As each context-free group language is actually deterministic context-free, it follows that all context-free groups are Church-Rosser groups. As the free abelian group of rank 2 is a non-context-free Church-Rosser group, this inclusion is proper. On the other hand, we show that there are co-context-free groups that are not growing context-sensitive. Also some closure and non-closure properties are established for the classes of Church-Rosser and growing context-sensitive groups.

More generally, we also establish some new characterizations and closure properties for the classes of Church-Rosser and growing context-sensitive languages. We use language theory to study the rational subset problem for groups and monoids.

## (PDF) The Homogeneous Spectrum of a Graded Commutative Ring | William Heinzer - faxarytogy.tk

We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for G-automata subsets of M.

It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership. We present an exposition of the theory of finite automata augmented with a multiply-only register storing an element of a given monoid or group. Included are a number of new results of a foundational nature. We illustrate our techniques with a group-theoretic interpretation and proof of a key theorem of Chomsky and Schutzenberger from formal language theory.

We study the commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Gamma is any graph not containing a four-cycle without chords, then the group G Gamma does not contain four elements whose commutation graph is a four-cycle; a consequence is that G Gamma does not have a subgroup isomorphic to a direct product of non-abelian free groups. We also obtain corresponding and more general results in the monoid case. We consider wreath product decompositions for semigroups of triangular matrices.

We exhibit an explicit wreath product decomposition for the semigroup of all n-by-n upper triangular matrices over a given field k, in terms of aperiodic semigroups and affine groups over k. In the case that k is finite this decomposition is optimal, in the sense that the number of group terms is equal to the group complexity of the semigroup.

We also obtain some decompositions for semigroups of triangular matrices over more general rings and semirings. We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which seem to be necessary to make the problem well-defined, the uniform word problem for semigroups described by automatic structures is decidable.

Under the same conditions, we show that one can also decide whether the semigroup is completely simple or completely zero-simple; in the case that it is, one can compute a Rees matrix representation for the semigroup, in the form of a Rees matrix together with an automatic structure for its maximal subgroup. On the other hand, we show that it is undecidable in general whether a given element of a given automatic monoid has a right inverse.

We calculate the spectra and spectral measures associated to random walks on restricted wreath products of finite groups with the infinite cyclic group, by calculating the Kesten-von Neumann-Serre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generalises the work of Grigorchuk and Zuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by automata coincides with the Kesten-von Neumann-Serre spectral measure. We consider blind, deterministic, finite automata equipped with a register which stores an element of a given monoid, and which is modified by right multiplication by monoid elements.

We show that, for monoids M drawn from a large class including groups, such an automaton accepts the word problem of a group H if and only if H has a finite index subgroup which embeds in the group of units of M. In the case that M is a group, this answers a question of Elston and Ostheimer. We consider the Krohn-Rhodes complexity of certain semigroups of upper triangular matrices over finite fields.

We consider the preservation of the properties of automaticity and prefix-automaticity in Rees matrix semigroups over semigroupoids and small categories. Some of our results are new or improve upon existing results in the single-object case of Rees matrix semigroups over semigroups. We consider various automata-theoretic properties of semigroupoids and small categories and their relationship to the corresponding properties in semigroups and monoids. We introduce natural definitions of finite automata and regular languages over finite graphs, generalising the usual notions over finite alphabets.

These allow us to introduce a definition of automaticity for semigroupoids and small categories, which generalises those introduced for semigroups by Hudson and for groupoids by Epstein.

We also introduce a definition of prefix-automaticity for semigroupoids and small categories, generalising that for certain monoids introduced by Silva and Steinberg. We study the relationship between automaticity properties in a semigroupoid and in a certain associated semigroup. This allows us to extend to semigroupoids and small categories a number of results about automatic and prefix-automatic semigroups and monoids.

In the course of our study, we also prove some new results about automaticity and prefix-automaticity in semigroups and monoids. These include the fact that prefix-automaticity is preserved under the taking of cofinite subsemigroups. Some of the material from the first February preprint of this paper now appears in a separate article, Automatic Rees matrix semigroups over categories , which can be found above.

We prove that any small cancellative category admits a faithful functor to a cancellative monoid. On the other hand a consequence of a recent result of Steinberg is that it is undecidable whether an ample semigroup embeds as a full subsemigroup of an inverse semigroup.

We begin with a brief introduction to the theory of word hyperbolic groups. We then consider four possible conditions which might reasonably be used as definitions or partial definitions of hyperbolicity in semigroups: having a hyperbolic Cayley graph; having hyperbolic Schutzenberger graphs; having a context-free multiplication table; or having word hyperbolic maximal subgroups.

Our main result is that these conditions coincide in the case of finitely generated completely simple semigroups. Final version not available in electronic form. Download an unrevised preprint postscript format. We extend various parts of the combinatorial theory of semigroups to encompass the closely related partial algebras of semigroupoids and small categories. We consider natural definitions of generators and relations for semigroupoids and hence for small categories. We associate to every semigroupoid a certain categorical-at-zero semigroup, and consider the relationship between presentations for the semigroupoid and the associated semigroup.

This allows us to extend to semigroupoids a number of important results concerning the properties of finite generation and finite presentability in semigroups. We also consider the preservation of these properties under Rees matrix constructions over semigroupoids. We introduce a definition of automaticity for semigroupoids and small categories, which generalises those introduced for semigroups by Hudson and for groupoids by Epstein. We consider the relationship between automaticity and prefix-automaticity in a semigroupoid and the associated categorical-at-zero semigroup.

This allows us immediately to extend to semigroupoids and small categories a number of results about automatic and prefix-automatic semigroups and monoids. We proceed to consider preservation of automaticity and prefix-automaticity under Rees matrix constructions over semigroupoids, obtaining some results which are new or improve upon existing results even in the semigroup single-object case.

We consider the relationship between the combinatorial properties of semigroupoids in general and semigroups in particular. We show that a semigroupoid is finitely generated [finitely presentable] exactly if the corresponding categorical-at-zero semigroup is finitely generated [respectively, finitely presentable]. This allows us to extend some of the main results of [Ruskuc ], to show that finite generation and presentability are preserved under finite extension of semigroupoids and the taking of cofinite semigroupoids.

This paper describes the definition and implementation of an OpenMP-like set of directives and library routines for shared memory parallel programming in Java. A specification of the directives and routines is proposed and discussed. A prototype implementation, consisting of a compiler and a runtime library, both written entirely in Java, is presented, which implements most of the proposed specification.

Some preliminary performance results are reported. This paper describes JOMP, a definition and implementation of a set of directives and library methods for shared memory parallel programming in Java. A specification of the OpenMP-like directives and methods is proposed.

A prototype implementation, consisting of a compiler and a runtime library both written entirely in Java is presented, which implements almost all of the proposed specification. Some preliminary performance results are also presented. A prototype implementation, JOMP, consisting of a compiler and a runtime library, both written entirely in Java, is presented, which implements a significant subset of the proposed specification.

OpenMP is a specification of directives and library routines for shared memory parallel programming. This report investigates the definition and implementation of OpenMP for Java, based on Java's native threads model. A specification for OpenMP for Java is proposed and discussed. A compiler and runtime library, both written entirely in Java, are presented, which together implement a large subset of the proposed specification. The concept of subtypes as sets of values of a given type has been used as a mathematical tool for reasoning about, and hence proving correctness of, functional programs.

Subtypes in functional programming are equivalent to the pre- and post- conditions used for formal specification in an imperative context.

We explore the possibility of representing such subtypes as functions expressable within a lazy functional language. These can then be incorporated into actual programs, providing extra verification of program correctness at run-time.

Module 17 - Fundamental Theorem of Finite Abelian Groups

Mark Kambites. Mark's Publications Where possible, links are given to final published versions. Tropical representations of plactic monoids with Marianne Johnson Preprint, June Download the preprint from the arXiv. Free objects in triangular matrix varieties and quiver algebras over semirings Preprint, April Linear isomorphisms preserving Green's relations for matrices over anti-negative semifields with Alexander Guterman and Marianne Johnson. Linear Algebra and its Applications , Vol.

• Passar bra ihop.
• Semigroups with finitely generated universal left congruence.
• A Clash of Kings (Song of Ice and Fire, Book 2).

On cogrowth, amenability and the spectral radius of a random walk on a semigroup with Robert Gray. To appear in International Mathematics Research Notices. Identities in upper triangular tropical matrix semigroups and the bicyclic monoid with Laure Daviaud and Marianne Johnson. Journal of Algebra Vol. Download the final version from the journal requires subscription of purchase. International Journal of Algebra and Computation Vol. Face monoid actions and tropical hyperplane arrangements with Marianne Johnson.

International Journal of Algebra and Computation. Pure dimension and projectivity of tropical polytopes with Zur Izhakian and Marianne Johnson. Finitely generated commutative monoids Finitely generated commutative monoids Finitely generated commutative monoids Finitely generated commutative monoids Finitely generated commutative monoids Finitely generated commutative monoids

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