Bicyclic Monoid Research Articles

Restrictions on local embeddability into finite semigroups

We expand the concept of local embeddability into finite structures (LEF) for the class of semigroups with investigations of non-LEF structures, a closely related generalising property of local wrapping of finite structures (LWF) and inverse semigroups. The established results include a description of a family of non-LEF semigroups unifying the bicyclic monoid and Baumslag–Solitar groups and demonstrating that inverse LWF semigroups with finite number of idempotents are LEF.

On topologization of the bicyclic monoid

On topologization of the bicyclic monoid

Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

We consider a tower of generalized rook monoid algebras over the field \mathbb{C} of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys–Murphy elements for generalized rook monoid algebras. Over an algebraically closed field \Bbbk of positive characteristic p , utilizing Jucys–Murphy elements of rook monoid algebras, for 0\leq i\leq p-1 we define the corresponding i -restriction and i -induction functors along with two extra functors. On the direct sum \mathcal{G}_{\mathbb{C}} of the Grothendieck groups of module categories over rook monoid algebras over \Bbbk , these functors induce an action of the tensor product of the universal enveloping algebra U(\widehat{\mathfrak{sl}}_p(\mathbb{C})) and the monoid algebra \mathbb{C}[\mathcal{B}] of the bicyclic monoid \mathcal{B} . Furthermore, we prove that \mathcal{G}_{\mathbb{C}} is isomorphic to the tensor product of the basic representation of U(\widehat{\mathfrak{sl}}_{p}(\mathbb{C})) and the unique infinite-dimensional simple module over \mathbb{C}[\mathcal{B}] , and also exhibit that \mathcal{G}_{\mathbb{C}} is a bialgebra. Under some natural restrictions on the characteristic of \Bbbk , we outline the corresponding result for generalized rook monoids.

Free objects in triangular matrix varieties and quiver algebras over semirings

We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all n×n upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative subsemigroups of quiver algebras over polynomial semirings. A case of particular interest is where n=2 and the semiring is the tropical semifield, where the variety coincides with that generated by the bicyclic monoid (or equivalently, by the free monogenic inverse monoid); here we are obtain faithful representations of the free objects in this variety inside a semidirect product of a commutative monoid acting on a semilattice. We apply these representations to answer several questions, including that of when the given varieties are locally finite.

On some generalization of the bicyclic monoid

On some generalization of the bicyclic monoid

Closed subsets of compact-like topological spaces

<p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of<br />countably pracompact topological spaces. We construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. This example provides an affirmative answer to a question posed by Banakh, Dimitrova, and Gutik in [4]. Also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a Hausdorff topological semigroup whose space is weakly H-closed.</p>

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

A Hausdorff topology τ on the bicyclic monoid with adjoined zero C0 is called weak if it is contained in the coarsest inverse semigroup topology on C0. We show that the lattice W of all weak shift-continuous topologies on C0 is isomorphic to the lattice SIF1×SIF1 where SIF1 is the set of all shift-invariant filters on ω with an attached element 1 endowed with the following partial order: F≤G if and only if G=1 or F⊂G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2c and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t.

Conjugacy in inverse semigroups

In a group G, elements a and b are conjugate if there exists g∈G such that g−1ag=b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a∼ib, if there exists g∈S1 such that g−1ag=b and gbg−1=a. The purpose of this paper is to study the conjugacy ∼i in several classes of inverse semigroups: symmetric inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid, stable inverse semigroups, and free inverse semigroups.

Geometry and algorithms for upper triangular tropical matrix identities

We provide geometric methods and algorithms to verify, construct and enumerate pairs of words (of specified length over a fixed m-letter alphabet) that form identities in the semigroup UTn(T) of n×n upper triangular tropical matrices. In the case n=2 these identities are precisely those satisfied by the bicyclic monoid, whilst in the case n=3 they form a subset of the identities which hold in the plactic monoid of rank 3. To each word we associate a signature sequence of lattice polytopes, and show that two words form an identity for UTn(T) if and only if their signatures are equal. Our algorithms are thus based on polyhedral computations and achieve optimal time complexity in some cases. For n=m=2 we prove a Structural Theorem, which allows us to quickly enumerate the pairs of words of fixed length which form identities for UT2(T). This allows us to recover a short proof of Adjan's theorem on minimal length identities for the bicyclic monoid, and to construct minimal length identities for UT3(T), providing counterexamples to a conjecture of Izhakian in this case. We conclude with six conjectures at the intersection of semigroup theory, probability and combinatorics, obtained through analysing the outputs of our algorithms.

The monoid of order isomorphisms between principal filters of $${\varvec{\mathbb {N}}}^n$$Nn

Let n be any positive integer and be the semigroup of all order isomorphisms between principal filters of the nth power of the set of positive integers $${\mathbb {N}}$$ with the product order. We study algebraic properties of the semigroup . In particular, we show that is a bisimple, E-unitary, F-inverse semigroup, describe Green’s relations on and its maximal subgroups. We show that the semigroup is isomorphic to the semidirect product of the direct nth power of the bicyclic monoid by the group of permutation . Also we prove that every non-identity congruence $${\mathfrak {C}}$$ on the semigroup is a group and describe the least group congruence on . We show that every Hausdorff shift-continuous topology on is discrete and discuss embedding of the semigroup into compact-like topological semigroups.

Identities in upper triangular tropical matrix semigroups and the bicyclic monoid

We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of n×n upper triangular tropical matrices, in terms of equivalence of certain tropical polynomials. This leads to an algorithm for checking whether such an identity holds, in time polynomial in the length of the identity and size of the alphabet. It also allows us to answer a question of Izhakian and Margolis, by showing that the identities which hold in the monoid of 2×2 upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid. Our results extend to a broader class of “chain structured tropical matrix semigroups”; we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup, which leads also to a representation by 3×3 upper triangular tropical matrices.

THE FINITE BASIS PROBLEM FOR THE MONOID OF TWO-BY-TWO UPPER TRIANGULAR TROPICAL MATRICES

For each positive $n$, let $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ denote the identity obtained from the Adjan identity $(xy)(yx)(xy)(xy)(yx)\approx (xy)(yx)(yx)(xy)(yx)$ by substituting $(xy)\rightarrow (x_{1}x_{2}\ldots x_{n})$ and $(yx)\rightarrow (x_{n}\ldots x_{2}x_{1})$. We show that every monoid which satisfies $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ for each positive $n$ and generates a variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid $U_{2}(\mathbb{T})$ (respectively, $U_{2}(\overline{\mathbb{Z}})$) of two-by-two upper triangular tropical matrices over the tropical semiring $\mathbb{T}=\mathbb{R}\cup \{-\infty \}$ (respectively, $\overline{\mathbb{Z}}=\mathbb{Z}\cup \{-\infty \}$) is nonfinitely based.

On surjunctive monoids

A monoid M is called surjunctive if every injective cellular automata with finite alphabet over M is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive.

Automata with Counters that Recognize Word Problems of Free Products

An M-automaton is a finite automaton with a blind counter that mimics a monoid M. The finitely generated groups whose word problems (when viewed as formal languages) are accepted by M-automata play a central role in understanding the family 𝔏(M) of all languages accepted by M-automata. If G1 and G2 are finitely generated groups whose word problems are languages in 𝔏(M), in general, the word problem of the free product G1 * G2 is not necessarily in 𝔏(M). However, we show that if M is enlarged to the free product M*P2, where P2 is the polycyclic monoid of rank two, then this closure property holds. In fact, we show more generally that the special word problem of M1 * M2 lies in 𝔏(M * P2) whenever M1 and M2 are finitely generated monoids with special word problems in 𝔏(M * P2). We also observe that there is a monoid without zero, denoted by CF2, that can be used in place of P2 for this purpose. The monoid CF2 is the rank two case of what we call a monoid with right invertible basis and its Rees quotient by its maximal ideal is P2. The fundamental theory of monoids with right invertible bases is completely analogous to that of free groups, and thus they are very convenient to use. We also investigate the questions of whether there is a group that can be used instead of the monoid P2 in the above result and under what circumstances P1 (or the bicyclic monoid) is enough to do the job of P2.

Identities of the plactic monoid

It is shown that the plactic monoid M of rank $$3$$ satisfies the identity $$wvvwvw=wvwvvw$$ where $$v=xyyx xy xyyx$$ and $$w= xyyx yx xyyx$$ . This is accomplished by first showing that certain simple monoids related to $$M$$ satisfy this identity. These simple monoids are natural generalizations of the bicyclic monoid $$B$$ , which satisfies the identity $$w=v$$ by a result of Adjan.

On sofic monoids

We investigate a notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.

Structure of Chinese algebras

The structure of the algebra K [ M ] of the Chinese monoid M over a field K is studied. The minimal prime ideals are described. They are determined by certain homogeneous congruences on M and they are in a one to one correspondence with diagrams of certain special type. There are finitely many such ideals. It is also shown that the prime radical B ( K [ M ] ) of K [ M ] coincides with the Jacobson radical and the monoid M embeds into the algebra K [ M ] / B ( K [ M ] ) . A new representation of M as a submonoid of the product B d × Z e for some natural numbers d, e, where B stands for the bicyclic monoid, is derived. Consequently, M satisfies a nontrivial identity.

Generalized Green'S Equivalences on the Subsemigroups of the Bicyclic Monoid

We study generalized Green's equivalences on all subsemigroups of the bicyclic monoid B and determine the abundant (and adequate) subsemigroups of B.

Semigroup identities in the monoid of two-by-two tropical matrices

We show that the monoid $M_{2}(\mathbb {T})$ of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity $$A^2B^4A^2A^2B^2A^2B^4A^2=A^2B^4A^2B^2A^2A^2B^4A^2.$$ Studying reduced identities for subsemigroups of $M_{2}(\mathbb {T})$ , and introducing a faithful semigroup representation for the bicyclic monoid by 2×2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.

Rational subsets of polycyclic monoids and valence automata

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, and the closure of the class of rational subsets under intersection and complement.