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A duoidal category is a category equipped with two monoidal structures in which one is (op)lax monoidal with respect to the other. In this paper we introduce duoidal ∞ -categories which are counterparts of duoidal categories in the setting of ∞ -categories. There are three kinds of functors between duoidal ∞ -categories, which are called bilax, double lax, and double oplax monoidal functors. We make three formulations of ∞ -categories of duoidal ∞ -categories according to which functors we take. Furthermore, corresponding to the three kinds of functors, we define bimonoids, double monoids, and double comonoids in duoidal ∞ -categories. [ABSTRACT FROM AUTHOR]

Partially ordered monoids (or pomonoids) S acting on a partially ordered set (or poset), briefly S -posets, appear naturally in the study of mappings between posets, and play an essential role in pomonoid theory. The study of flatness properties of S -posets was initiated by Fakhruddin in the 1980s, and an extensive theory of flatness properties has been developed in the past several decades. The obtained results have prompted a new progress in the research area of S -posets. To date, a large number of familiar properties have been generalized from acts to S -posets (involving free and projective S -posets, flat S -posets of various sorts, S -posets satisfying Conditions (P) , (W P) and (P W P) , and torsion free S -posets). Some new properties in S -posets, such as Conditions (P w) , (W P) w and (P W P) w , have also been discovered. This paper continues the study of flatness properties of S -posets. We first introduce Condition (P ′) in the context of S -posets, and characterize pomonoids by this new property of S -posets. Unlike the case for acts, pomonoids over which all right S -posets satisfy Condition (P ′) are stronger than pogroups. Thereby, we introduce Condition (P w ′) similar to Condition (P w). Furthermore, we describe Conditions (P ′) and (P w ′) covers of cyclic S -posets. Finally, we investigate direct products of S -posets satisfying Conditions (P ′) and (P w ′). [ABSTRACT FROM AUTHOR]

AbstractThis paper aims to investigate a new generalization of injective S-acts, leading us to study when a subact of a projective S-act remains projective. Furthermore, we consider injective S-acts relative to all embeddings from (finitely generated) S-acts to a fixed S-act A, called (finitely) A-injective. Then, we introduce the concept of FF-injectivity, which lies between injectivity and absolute purity, defined as injectivity relative to all embeddings from finitely generated S-acts to finitely generated S-acts. Additionally, the relationship between solutions of consistent sets of equations and FF-injectivity is considered. Finally, classifications of monoids by the properties of these types of injective acts are discussed. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we show how the algorithms to explore the tree of numerical semigroups can be used to calculate the winningness of positions in the game of Sylver Coinage. We introduce a new invariant for numerical semigroups, the minimal distance, and show how it can be used to prune the tree of numerical semigroups for an efficient way of calculating the winningness of positions in the game of Sylver Coinage. We end with a few open questions that were spawned as a result of this research. [ABSTRACT FROM AUTHOR]

A variety of algebras is called Cross if it is finitely based, finitely generated, and has finitely many subvarieties. In this paper, we classify all Cross varieties of aperiodic monoids with commuting idempotents. [ABSTRACT FROM AUTHOR]

This paper determines relations between two notions concerning monoids: factorability structure, introduced to simplify the bar complex; and quadratic normalization, introduced to generalize quadratic rewriting systems and normalizations arising from Garside families. Factorable monoids are characterized in the axiomatic setting of quadratic normalizations. Additionally, quadratic normalizations of class 4 , 3 are characterized in terms of factorability structures and a condition ensuring the termination of the associated rewriting system. [ABSTRACT FROM AUTHOR]

Every non-erasing monoid morphism σ : A ∗ → B ∗ induces a measure transfer map σ X M : M (X) → M (σ (X)) between the measure cones M (X) and M (σ (X)) , associated to any subshift X ⊆ A Z and its image subshift σ (X) ⊆ B Z respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective (Bédaride et al 2024 Ergod. Theor. Dynam. Syst. 44 3120–54). Furthermore, an efficient technique to compute the value of the transferred measure σ X M (μ) on any cylinder [w] (for w ∈ B ∗ ) is presented. Theorem. If a non-erasing morphism σ : A ∗ → B ∗ is injective on the shift-orbits of some subshift X ⊆ A Z , then σ X M is injective. The assumption on σ that it is 'injective on the shift-orbits of X ' is strictly weaker than 'recognizable in X ', and strictly stronger than 'recognizable for aperiodic points in X '. The last assumption does in general not suffice to obtain the injectivity of the measure transfer map σ X M . [ABSTRACT FROM AUTHOR]

An algebra that generates a variety with uncountably many subvarieties is said to be of type 2 ℵ 0 . We show that the Rees quotient monoid M(aabb) of order ten is of type 2 ℵ 0 , thereby affirmatively answering a recent question of Glasson. As a corollary, we exhibit a new example of type 2 ℵ 0 monoid of order six, which turns out to be the minimal possible cardinality and the first of its kind that is finitely based. [ABSTRACT FROM AUTHOR]

We consider monoids equipped with a compatible quantale valued relation, which we call quantale enriched monoids, and study semidirect products of such structures. It is well-known that semidirect products of monoids are closely related to Schreier split extensions which, in the setting of monoids, play the role of split extensions of groups. We will thus introduce certain split extensions of quantale enriched monoids, which generalize the classical Schreier split extensions of monoids, and investigate their connections with semidirect products. We then restrict our study to a class of quantale enriched monoids whose behavior mimics the fact that the preorder on a preordered group is completely determined by its cone of positive elements. Finally, we instantiate our results for preordered monoids and compare them with existing literature. [ABSTRACT FROM AUTHOR]

The problem of determining the size of a minimal generating set (the rank) of a semigroup has both a combinatorial and algebraic nature. For example, the rank of full transformation semigroup and the partition monoid of degree n eventually does not depend on n, however, the rank of the semigroup of binary relations B n increases exponentially with n. In this paper, we show that the rank of the semigroup of partitioned binary relations P B n also increases exponentially with n. [ABSTRACT FROM AUTHOR]

A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the question of whether the fact that a torsion-free monoid M and an integral domain R are both atomic implies that the monoid algebra R [ M ] of M over R is also atomic. In general this is not true, and the first negative answer to this question was given by Roitman in 1993: he constructed an atomic integral domain whose polynomial extension is not atomic. More recently, Coykendall and the first author constructed finite-rank torsion-free atomic monoids whose monoid algebras over certain finite fields are not atomic. Still, the ascent of atomicity from finite-rank torsion-free monoids to their corresponding monoid algebras over fields of characteristic zero is an open problem. Coykendall and the first author also constructed an infinite-rank torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. As the primary result of this paper, we construct a rank-one torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. To do so, we introduce and study a methodological construction inside the class of rank-one torsion-free monoids that we call lifting, which consists in embedding a given monoid into another monoid that is often more tractable from the arithmetic viewpoint. For instance, we prove here that the embedding in the lifting construction preserves the ascending chain condition on principal ideals and the existence of maximal common divisors. [ABSTRACT FROM AUTHOR]

We show that any graph product of residually finite monoids is residually finite. As a special case we obtain that any free product of residually finite monoids is residually finite. The corresponding results for graph products of semigroups follow. [ABSTRACT FROM AUTHOR]

The study of flat ring morphisms is an important theme in commutative algebra. The purpose of this article is to develop an abstract theory of flatness in the framework of coherent quantales. The first question we must address is the definition of a notion of "flat quantale morphism" as an abstraction of flat ring morphisms. For this, we start from a characterization of the flat ring morphism in terms of the ideal residuation theory. The flat coherent quantale morphism is studied in relation to the localization of coherent quantales. The quantale generalizations of some classical theorems from the flat ring morphisms theory are proved. The Going-down and Going-up properties are then studied in connection with localization theory and flat quantale morphisms. As an application, characterizations of zero-dimensional coherent quantales are obtained, formulated in terms of Going-down, Going-up, and localization. We also prove two characterization theorems for the coherent quantales of dimension at most one. The results of the paper can be applied both in the theory of commutative rings and to other algebraic structures: F-rings, semirings, bounded distributive lattices, commutative monoids, etc. [ABSTRACT FROM AUTHOR]

AbstractUsing techniques of algebraic and analytic number theory, we resolve a question on monoid rings posed by Kulosman et al. under the assumption of the Generalized Riemann Hypothesis (GRH). Specifically, we show that under an appropriate GRH, for any (rational) prime p the set E(p)={q prime|Xq−1 factors in Fp[X;M]}, where M=〈2,3〉=N0∖{1}, contains a subset with positive natural density. In particular E(p)≠∅. This proves that M is not a so-called “Matsuda monoid” of any positive type. For p=2,3 this was observed by Kulosman, who provided factorizations of X7−1 and X11−1 in F2[X;M] and F3[X;M], respectively. Our results explain and reproduce both of these factorizations, as well. [ABSTRACT FROM AUTHOR]

A submonoid of the additive group ℚ is called a Puiseux monoid if it consists of non-negative rationals. Given a monoid M, the set consisting of all non-empty finite subsets of M is also a monoid under the Minkowski sum, and it is called the (finitary) power monoid of M. In this paper, we study atomicity and factorization properties in the setting of power monoids of Puiseux monoids. We specifically focus on the ascent of atomicity and the finite factorization property (although we also consider the ascending chain condition on principal ideals and the length-finite factorization property). We prove that the finite factorization property ascends from any Puiseux monoid to its power monoid. On the other hand, we construct an atomic Puiseux monoid whose power monoid is not atomic. We also prove that the existence of maximal common divisors for non-empty finite subsets is a sufficient condition for the property of being atomic to ascend from a Puiseux monoid to its power monoid. [ABSTRACT FROM AUTHOR]

A gap a of a numerical semigroup S is fundamental if { 2 a , 3 a } ⊆ S. In this work, we will study the set B (a) = S ∣ S is a numerical semigroup and a is a fundamental gap of S. In particular, we will give an algorithm to compute all the elements of B (a) with a given genus. The intersection of two elements of B (a) is again one element of B (a). A B (a) -irreducible numerical semigroup is an element of B (a) that cannot be expressed as an intersection of two elements of B (a) containing it properly. In this paper, we will study the B (a) -irreducible numerical semigroups. In this sense we will give an algorithm to calculate all of them. Finally, we will study the submonoids of (N , +) that can be expressed as an intersection (finite or infinite) of elements belonging to B (a). [ABSTRACT FROM AUTHOR]

We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity classification for the same problem over groups. [ABSTRACT FROM AUTHOR]

We introduce the basic notions and present examples and results on Lie categories - categories internal to the category of smooth manifolds. Demonstrating how the units of a Lie category C dictate the behavior of its invertible morphisms G.C/, we develop sufficient conditions for G.C/ to form a Lie groupoid. We show that the construction of Lie algebroids from the theory of Lie groupoids carries through, and ask when the Lie algebroid of G.C/ is recovered. We reveal that the lack of the invertibility assumption on morphisms leads to a natural generalization of rank from linear algebra, develop its general properties, and show how the existence of an extension C! G of a Lie category to a Lie groupoid affects the ranks of morphisms and the algebroids of C. Furthermore, certain completeness results for invariant vector fields on Lie monoids and Lie categories with well-behaved boundaries are obtained. Interpreting the developed framework in the context of physical processes, we yield a rigorous approach to the theory of statistical thermodynamics by observing that entropy change, associated to a physical process, is a functor. [ABSTRACT FROM AUTHOR]

We study the four plactic-like monoids that arise by taking the meets and joins of stalactic and taiga congruences. We obtain the combinatorial objects associated with the meet monoids, establishing Robinson–Schensted-like correspondences and giving extraction and iterative insertion algorithms for these objects. We then obtain results on the sizes of classes of words equal in plactic-like monoids, show that some of these monoids are syntactic, and characterise their equational theories. [ABSTRACT FROM AUTHOR]

We say that a topological monoid S is left non-archimedean (in short: l-NA) if the left action of S on itself admits a proper S-compactification ν : S ↪ Y such that Y is a Stone space. This provides a natural generalization of the well known concept of NA topological groups. The Stone and Pontryagin dualities play a major role in achieving useful characterizations of NA monoids. We show that many naturally defined topological monoids are NA and present universal NA monoids. Among others, we prove that the Polish monoid C (2 ω , 2 ω) is a universal separable metrizable l-NA monoid and the Polish monoid N N is universal for separable metrizable r-NA monoids. [ABSTRACT FROM AUTHOR]

We prove that a plactic monoid of any finite rank has decidable first order theory. This resolves other open decidability problems about the finite rank plactic monoids, such as the Diophantine problem and identity checking. This is achieved by interpreting a plactic monoid of arbitrary rank in Presburger arithmetic, which is known to have decidable first order theory. We also prove that the interpretation of the plactic monoids into Presburger Arithmetic is in fact a bi-interpretation, hence any two plactic monoids of finite rank are bi-interpretable with one another. The algorithm generating the interpretations is uniform, which answers positively the decidability of the Diophantine problem for the infinite rank plactic monoid. [ABSTRACT FROM AUTHOR]

A generalization to the categorical notion of biproduct, called semibiproduct, which in the case of groups covers classical semidirect products, has recently been analysed in the category of monoids with surprising results in the classification of weakly Schreier extensions. The purpose of this paper is to extend the study of semibiproducts to the category of semigroups. However, it is observed that a further analysis into the category of magmas is required in attaining a full comprehension on the subject. Indeed, although there is a subclass of magma-actions, that we call representable, which classifies all semibiproducts of magmas whose behaviour is similar to semigroups, it is nevertheless more general than the subclass of associative magma-actions. [ABSTRACT FROM AUTHOR]

Monoids generated by elements of order two appear in numerous places in the literature. For example, Coxeter reflection groups in geometry, Kuratowski monoids in topology, various monoids generated by regular operations in language theory and so on. In order to initiate a classification of these monoids, we are interested in the subproblem of monoids, called strict Projection Involution Monoids (2-PIMs), generated by an involution and an idempotent. In this case we show, when the monoid is finite, that it is generated by a single equation (in addition to the two defining the involution and the idempotent). We then describe the exact possible forms of this equation and classify them. We recover Kuratowski's theorem as a special case of our study. [ABSTRACT FROM AUTHOR]

In this paper we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to determine their ranks. We also describe their Green's relations, calculate their cardinalities and study their regularity. [ABSTRACT FROM AUTHOR]

AbstractInteractive data visualization has become a staple of modern data presentation. Yet, despite its growing popularity, we still lack a general framework for turning raw data into summary statistics that can be displayed by interactive graphics. This gap may stem from a subtle yet profound issue: while we would often like to treat graphics, statistics, and interaction in our plots as independent, they are in fact deeply connected. This article examines this interdependence in light of two fundamental concepts from category theory: groups and monoids. We argue that the knowledge of these algebraic structures can help us design sensible interactive graphics. Specifically, if we want our graphics to support interactive features which split our data into parts and then combine these parts back together (such as linked selection), then the statistics underlying our plots need to possess certain properties. By grounding our thinking in these algebraic concepts, we may be able to build more flexible and expressive interactive data visualization systems. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

The category STROPm of supertropical monoids, whose morphisms are transmissions, has the full-reflective subcategory STROP of commutative semirings. In this setup, quotients are determined directly by equivalence relations, as ideals are not applicable for monoids, leading to a new approach to factorization theory. To this end, tangible factorization into irreducibles is obtained through fiber contractions and their hierarchy. Fiber contractions also provide different quotient structures, associated with covers and types of splitting covers. [ABSTRACT FROM AUTHOR]

This paper studies groups and their coset monoids. The anti-abnormal subgroups of a group are firstly introduced and investigated. It is shown that a group is an N ˜ -group if and only if each subgroup of it is anti-abnormal. Also, it is proved that the coset semigroups of N ˜ -groups are exactly the E-reflexive inverse semigroups which are factorisable and the natural connection between their semilattice of idempotents and lattice of subgroups of their group of units is a dual isomorphism. Finally, some characterizations of the coset semigroups of S-groups (U-groups and residually central groups respectively) are given. This extends McAlister's result in 1980. [ABSTRACT FROM AUTHOR]

Let A ⊆ B be commutative rings with unity, let S ⊆ T be torsion-free cancellative monoids, and let n ≥ 1 . We give a characterization of when the monoid ring A [ S ] is n-root closed in the monoid ring B [ T ] . For torsion-free cancellative ordered monoids (S , ≤) ⊆ (T , ≤) , we also present sufficient conditions and necessary conditions for the generalized power series ring [ ​ [ A S , ≤ ] ​ ] to be n-root closed in the generalized power series ring [ ​ [ B T , ≤ ] ​ ] . [ABSTRACT FROM AUTHOR]

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus T q , the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of T q ; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals. [ABSTRACT FROM AUTHOR]

An additive submonoid of the nonnegative cone of the real line is called a positive monoid. Positive monoids consisting of rational numbers (also known as Puiseux monoids) have been the subject of several recent papers. Moreover, those generated by a geometric sequence have also received a great deal of recent attention. Our purpose is to survey many of the recent advances regarding positive monoids, and we provide numerous examples to illustrate the complexity of their atomic and arithmetic structures. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the structure of semilattices of monoids via structural homomorphisms, with special attention to semilattices of unipotent (one idempotent) monoids. This is an extension of the well-known construction of Clifford semigroups (semilattices of groups). It is similar to the construction of a right abundant semigroup with central idempotents, characterized by Fountain as a strong semilattice of left cancellative monoids. [ABSTRACT FROM AUTHOR]

We study Hopf classes: families of pure ordered simplicial complexes that give rise to Hopf monoids under join and deletion/contraction. The prototypical Hopf class is the family of ordered matroids. The idea of a Hopf class leads to a systematic study of simplicial complexes related to matroids, including shifted complexes and broken-circuit complexes. We compute the Hopf antipodes in two cases: facet-initial complexes (which generalize shifted complexes) and unbounded ordered matroids. The latter calculation uses the topological method of Aguiar and Ardila, complicated by certain auxiliary simplicial complexes that we call Scrope complexes, whose Euler characteristics control the coefficients of the antipode. The resulting antipode formula is multiplicity-free and cancellation-free. [ABSTRACT FROM AUTHOR]

Given two numerical semigroups S and T we say that T is a multiple of S if there exists an integer d ∈ N \ {0} such that S = {x ∈ N | dx ∈ T}. In this paper we study the family of multiples of a (fixed) numerical semigroup. We also address the open problem of finding numerical semigroups of embedding dimension e without any quotient of embedding dimension less than e, and provide new families with this property. [ABSTRACT FROM AUTHOR]

In this paper, we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids wEndCn and EndCn of all weak endomorphisms and all endomorphisms of an undirected cycle graph Cn with n vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities. [ABSTRACT FROM AUTHOR]

Recently, Izhakian and Merlet gave a faithful representation ρ ~ of the Chinese monoid C h n of every finite rank n as a submonoid of the monoid U T 2 · 3 n - 2 (T) of upper triangular matrices over the tropical semiring T . We exhibit another faithful representation ϕ ~ n of C h n as a submonoid of the monoid U T n (n - 1) (T) of upper triangular matrices over T . The dimension of ϕ ~ n is smaller than that of ρ ~ when n ⩾ 4 . Further, we give a faithful representation of the Chinese monoid (C h n , ♯) under Schützenberger's involution ♯ . [ABSTRACT FROM AUTHOR]

We describe the mutation class of a certain quiver with a frozen vertex and associate these quivers with potentials appearing in our mutation class to presentations of the monoid P n + 1 of uniform block permutations on the set { 1 , 2 , ... , n + 1 } . Some classical and known presentations of P n + 1 , including FitzGerald's presentation and Everitt and Fountain's presentation, are recovered. [ABSTRACT FROM AUTHOR]

An algebra is finitely related (or has finite degree) if its term functions are determined by some finite set of finitary relations. Nilpotent monoids built from words, via Rees quotients of free monoids, have been used to exhibit many interesting properties with respect to the finite basis problem. We show that much of this intriguing behaviour extends to the world of finite relatedness by using interlocking patterns called chain, crown, and maelstrom words. In particular, we show that there are large classes of non-finitely related nilpotent monoids that can be used to construct examples of: ascending chains of varieties alternating between finitely and non-finitely related; non-finitely related semigroups whose direct product are finitely related; the addition of an identity element to a non-finitely related semigroup to produce a finitely related semigroup. [ABSTRACT FROM AUTHOR]

Let C ⊂ N p be an integer polyhedral cone. An affine semigroup S ⊂ C is a C -semigroup if | C ∖ S | < + ∞ . This structure has always been studied using a monomial order. The main issue is that the choice of these orders is arbitrary. In the present work, we choose the order given by the semigroup itself, which is a more natural order. This allows us to generalise some of the definitions and results known from numerical semigroup theory to C -semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we give a sufficient condition under which an involution monoid generates a variety with continuum many subvarieties. According to this result, several involution J -trivial monoids are shown to generate varieties with continuum many subvarieties. These examples include Rees quotients of free involution monoids, Lee monoids with involution, and Straubing monoids with involution. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function. [ABSTRACT FROM AUTHOR]

After one‐point compactification, the collection of all unordered configuration spaces of a manifold admits a commutative multiplication by superposition of configurations. We explain a simple (derived) presentation for this commutative monoid object. Using this presentation, one can quickly deduce Knudsen's formula for the rational cohomology of configuration spaces, prove rational homological stability and understand how automorphisms of the manifold act on the cohomology of configuration spaces. Similar considerations reproduce the work of Farb–Wolfson–Wood on homological densities. [ABSTRACT FROM AUTHOR]

Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of product and permutation categories (PROPs) and product and braid categories (PROBs) with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action. [ABSTRACT FROM AUTHOR]

We generalize free monoids by defining k-monoids. These are nothing other than the one-vertex higher-rank graphs used in C ∗ -algebra theory with the cardinality requirement waived. The 1-monoids are precisely the free monoids. We then take the next step and generalize k-monoids in such a way that self-similar group actions yield monoids of this type. [ABSTRACT FROM AUTHOR]

The plactic monoids can be obtained from the tensor product of crystals. Similarly, the hypoplactic monoids can be obtained from the quasi-tensor product of quasi-crystals. In this paper, we present a unified approach to these constructions by expressing them in the context of quasi-crystals. We provide a sufficient condition to obtain a quasi-crystal monoid for the quasi-tensor product from a quasi-crystal monoid for the tensor product. We also establish a sufficient condition for a hypoplactic monoid to be a quotient of the plactic monoid associated to the same seminormal quasi-crystal. [ABSTRACT FROM AUTHOR]

We study the equational theories and bases of meets and joins of several varieties of plactic-like monoids. Using those results, we construct sublattices of the lattice of varieties of monoids, generated by said varieties. We calculate the axiomatic ranks of their elements, obtain plactic-like congruences whose corresponding factor monoids generate varieties in the lattice, and determine which varieties are joins of the variety of commutative monoids and a finitely generated variety. We also show that the hyposylvester and metasylvester monoids generate the same variety as the sylvester monoid. [ABSTRACT FROM AUTHOR]

Let (프n,ρ) be the involution monoid of Annular monoid 프n under the rotation involution ρ. The involution monoids (프1,ρ) and (프2,ρ) are easily seen to be finitely based; Auinger et al. proved that (프n,ρ) is inherently non-finitely based if n ≥ 4. In this paper, we show that (프3,ρ) is finitely based by providing a finite identity basis for (프3,ρ), which answers an open question posed by Auinger et al. Therefore, the involution monoid (프n,ρ) is finitely based if and only if n ≤ 3. [ABSTRACT FROM AUTHOR]