Coset Decomposition of Positively Self-Conjugate Inverse Submonoids of Polycyclic Monoids

Abstract. We obtain several new results about the coding of a monoid by an appropriate submonoid of a polycyclic monoid. In particular, we characterize groups, periodic monoids, and right cancellative monoids in terms of the coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids.     

On presentations of Brauer-type monoids

We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.     

Coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids

We obtain several new results about the coding of a monoid by an appropriate submonoid of a polycyclic monoid. In particular, we characterize groups, periodic monoids, and right cancellative monoids in terms of the coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids.     

Inverse Subsemigroups of the Monogenic Free Inverse Semigroup

It is shown that every finitely generated inverse subsemigroup (submonoid) of the monogenic free inverse semigroup (monoid) is finitely presented. As a consequence, the homomorphism and the isomorphism problems for the monogenic free inverse semigroup (monoid) are proven to be decidable.     

The monoid of ordered partitions of a natural number

The sequences of non-negative integers form a monoid, a natural submonoid of which has elements corresponding to order-preserving transformations of a finite chain. This, in turn, has a submonoid whose elements are ordered partitions of a natural number. A presentation for the last monoid is given, and the inclusion poset of principal right ideals is described. The poset of principal left ideals has a recursive structure that gives rise to an interesting sequence of numbers.     

Equivaleces between D-categories of Inverse Monoids

There exist two equivalent small categories associated to an inverse monoid which reflect its divisorial structure, called its D-categories. It is well known the relationship between the Green's relation D and the structure, of certain subsemigroups, of an inverse monoid. In this work, an analogous result is established between the Green's relation J and the D-categories of those subsemigroups. They are also given equivalent conditions for the equivalence of the D-categories of two inverse monoids and likewise equivalent conditions for the isomorphism of two inverse monoids in terms of its D-categories. It is proved that for many important classes of inverse monoids the multiplicative structure is determined by the associated category. On the contrary, a sufficient condition to obtain families of counterexamples to the above is provided and three examples are explicitely exhibited.     

Periodic subsemigroups of endomorphism monoids

If S is a periodic subsemigroup of the endomorphism monoid of a polycyclic group, then Endimioni (Mediterr J Math 8:307–313, 2011) proved that S is locally finite. Here we present an alternative proof that also extends the result to groups with suitable rank restrictions. Further we give an alternative proof of McNaughton and Zalcstein’s (J Algebra 34:292–299, 1975) theorem that periodic multiplicative subsemigroups of a matrix ring over a field are also locally finite. Finally we extend the latter to periodic subsemigroups of the endomorphism ring of a finitely generated module over a commutative ring.     

Ordered groupoid quotients and congruences on inverse semigroups

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s \({\mathcal {J}}\)–relation. The corresponding equivalence relation \(\simeq _N\) is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.     

Structure of a finite commutative inverse semigroup and a finite bundle for which the inverse monoid of local automorphisms is permutable

For a semigroup S, the set of all isomorphisms between the subsemigroups of the semigroup S with respect to composition is an inverse monoid denoted by PA(S) and called the monoid of local automorphisms of the semigroup S. The semigroup S is called permutable if, for any couple of congruences ρ and σ on S, we have ρ ∘ σ = σ ∘ ρ. We describe the structures of a finite commutative inverse semigroup and a finite bundle whose monoids of local automorphisms are permutable.     

Strong representations of the polycyclic inverse monoids: Cycles and atoms

This paper was inspired by a monograph by Bratteli and Jorgensen, and the work of Kawamura. We introduce two new semigroups: a wide inverse submonoid of the polycyclic inverse monoid, called the gauge inverse monoid, and a Zappa-Szép product of an arbitrary free monoid with the free monoid on one generator. Both these monoids play an important role in studying arbitrary, not necessarily transitive, strong actions of polycyclic inverse monoids. As a special case of such actions, we obtain some new results concerning the strong actions of P ₂ on ℤ determined by the choice of one positive odd number. We explain the role played by Lyndon words in characterising these repesentations and show that the structure of the representation can be explained by studying the binary representations of the numbers $\frac{1} {p},\frac{2} {p}, \ldots \frac{{p - 1}} {p}$\frac{1} {p},\frac{2} {p}, \ldots \frac{{p - 1}} {p}. We also raise some questions about strong representations of the polycyclic monoids on free abelian groups.     

An order topologyfor finitely generated free monoids

In this article a method is given for embedding a finitely generated free monoid as a dense subset of the unit interval. This gives an order topology for the monoid such that the submonoids generated by an important class of maximal codes occur as “thick” subsets. As an ordered topological space, the notion of thickness in a frec monoid can be interpreted in a number of ways. One such notion is that of density. In particular, subsets of a free monoid that fail to meet all two sided ideals (the thin sets, of which recognizable codes are an example) are shown (corollary 4.2) to be nowhere dense. Furthermore, it is shown (corollary 5.1) that a thin code is maximal if and only if the submonoid that it generates is dense on some interval. Thus thin codes that are maximal are precisely those that generate thick submonoids. Another notion of thickness is that of category. The embedding allows the free monoid to be viewed as a subspace of the unit interval. In theorem 5.6 it is shown that a thin code is maximal just in case the closure of the submonoid that it generates is second category in the unit interval. A mild connection with Lebesque measure is then made. In what follows, all free monoids are assumed to be generated by a finite set of at least two elements. IfA is such a set, thenA ^* denotes the free monoid generated byA. The setA is called an alphabet, the elements ofA ^* are called words, ande denotes the empty word inA ^*. Topological terminology and notation follows that of Kelley [2].     

A Note on Almost GCD Monoids

A commutative cancellative monoid H (with 0 adjoined) is called an almost GCD (AGCD) monoid if for x,y in H, there exists a natural number n = n(x,y) so that xⁿ and yⁿ have an LCM, that is, xⁿH \cap yⁿH is principal. We relate AGCD monoids to the recently introduced inside factorial monoids (there is a subset Q of H so that the submonoid F of H generated by Q and the units of H is factorial and some power of each element of H is in F). For example, we show that an inside factorial monoid H is an AGCD monoid if and only if the elements of Q are primary in H, or equivalently, H is weakly Krull, distinct elements of Q are v-coprime in H, or the radical of each element of Q is a maximal t-ideal of H. Conditions are given for an AGCD monoid to be inside factorial and the results are put in the context of integral domains.     

相对于幺半群的McCoy环的扩张

对于幺半群~$M$, 本文引入了~$M$-McCoy~环.~证明了~$R$~是~$M$-McCoy~环当且仅当~$R$~上的~$n$~阶上三角矩阵环~$aUT_n(R)$~是~$M$-McCoy~环;得到了若~$R$~是~McCoy~环,~$R[x]$~是~$M$-McCoy~环,则~$R[M]$~是~McCoy~环;对于包含无限循环子半群的交换可消幺半群~$M$,证明了若~$R$~是~$M$-McCoy~环,则半群环~$R[M]$~是~McCoy~环及~$R$~上的多项式环~$R[x]$~是~$M$-McCoy~环.     

Idempotent-Conjugate Monoids with Nilpotent Unit Groups

Let M be a finite monoid with unit group G such that J-related idempotents in M are conjugate. If G is nilpotent, we prove that the complex monoid algebra CM of M is semisimple if and only if M is an inverse monoid. Conversely let G be a finite group such that for any finite idempotent-conjugate monoid M with unit group G, CM semisimple implies that M is an inverse monoid. We then show that G is a nilpotent group.     

Fixed points of endomorphisms of trace monoids

It is proved that the fixed point submonoid and the periodic point submonoid of a trace monoid endomorphism are always finitely generated. If the dependence alphabet is a transitive forest, it is proved that the set of regular fixed points of the (Scott) continuous extension of an endomorphism to real traces is Ω-rational for every endomorphism if and only if the monoid is a free product of free commutative monoids.     

On finitely presented,cancellative and commutative ordered monoids

We show that every finitely presented, cancellative and commutative ordered monoid is determined by a finitely generated and cancellative pseudoorder on the monoid (ℕ ⁿ ,+) for some positive integer n. Every cancellative pseudoorder on (ℕ ⁿ ,+) is determined by a submonoid of the group (ℤ ⁿ ,+), and we prove that the pseudoorder is finitely generated if and only if the submonoid is an affine monoid in ℤ ⁿ .     

Fundamental groups,inverse schützenberger automata,and monoid presentations

This paper gives decidable conditions for when a finitely generated subgroup of a free group is the fundamental group of a Schützenberger automaton corresponding to a monoid presentation of an inverse monoid. Also, generalizations are given to specific types of inverse monoids as well as to monoids which are "nearly inverse." This result has applications to computing membership for inverse monoids in a Mal'cev product of the pseudovariety of semilattices with a pseudovariety of groups.

This paper also shows that there is a bijection between strongly connected inverse automata and subgroups of a free group, generated by positive words. Hence, we also obtain that it is decidable whether a finite strongly connected inverse automaton is a Schützenberger automaton corresponding to a monoid presentation of an inverse monoid. Again, we have generalizations to other types of inverse monoids and to "nearly inverse" monoids. We show that it is undecidable whether a finite strongly connected inverse automaton is a Schützenberger automaton of a monoid presentation of anE-unitary inverse monoid.     

Inverse subsemigroups and classes of finite aperiodic semigroups

We prove a number of results related to finite semigroups and their inverse subsemigroups, including the following. (1) A finite semigroup is aperiodic if and only if it is a homomorphic image of a finite semigroup whose inverse subsemigroups are semilattices. (2) A finite inverse semigroup can be represented by order-preserving mappings on a chain if and only if it is a semilattice. Finally, we introduce the concept of pseudo-small quasivariety of finite semigroups, generalizing the concept of small variety.