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A centralizing monoid M is a set of unary operations which commute with some set F of operations. Here, F is called a witness of M. On a 3-element set, a centralizing monoid is maximal if and only if it has a constant operation or a majority minimal operation as its witness. In this paper, we take one such majority operation, which corresponds to a maximal centralizing monoid, on a 3-element set and obtain its generalization, called mb, on a k-element set for any k > 3. We explicitly describe the centralizing monoid M(mb) with mb as its witness and then prove that it is not maximal if k > 3, contrary to the case for k = 3.

We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to show that the 9-element monoid L41 is non-finitely based. The monoid L41 was the only unsolved case in the finite basis problem for Lee monoids Lℓ1, obtained by adjoining an identity element to the semigroup Lℓ generated by two idempotents a and b subjected to the relation 0=abab⋯ (length ℓ). We also prove a syntactic sufficient condition which is equivalent to the sufficient condition of Lee under which a semigroup is non-finitely based. This gives a new proof to the results of Zhang–Luo and Lee that the semigroup Lℓ is non-finitely based for each ℓ≥3.

In this paper, we study the atomic structure of Puiseux monoids generated by monotone sequences. To understand this atomic structure, it is often useful to know whether the monoid has a bounded generating set. We provide necessary and sufficient conditions for the atomicity and boundedness to be transferred from a monotone Puiseux monoid to all its submonoids. Finally, we present two special subfamilies of monotone Puiseux monoids and fully classify their atomic structure.

For factorizing representative (or rational) series, with coefficients in a commutative ring A containing ℚ, we examine various products such as concatenation, shuffle and its ϕ -deformations, … (and their co-products) defined on the free monoid which are such that their associated bialgebras are isomorphic to the Sweedler’s dual, for A being a field K .

It is shown that, for every prime number p, the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gp of all finite p-groups has the cardinality of the continuum. Furthermore, it is shown, in addition, that the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gsol of all finite solvable groups has also the cardinality of the continuum.

We exhibit explicit and easily realisable bijections between Hecke–Kiselman monoids of type An/A~n; certain braid diagrams on the plane/cylinder; and couples of integer sequences of particular types. This yields a fast solution of the word problem and an efficient normal form for these HK monoids. Yang–Baxter type actions play an important role in our constructions.

We define an algebraic structure similar to that of a semiring, but without some of the requirements. As it is somehow also similar to the structure of left brace, we call it an M -brace. We present a connection between Garside monoids and more generally lcm-monoids with this algebraic structure. An lcm-monoid M is a left-cancellative monoid such that 1 is the unique invertible element in M , and every pair of elements in M admit an lcm with respect to left-divisibility. The class of lcm-monoids contains the Gaussian, quasi-Garside and Garside monoids.

We observe that for each n≥2 , the identities of the stylic monoid with n generators coincide with the identities of n-generated monoids from other distinguished series of J -trivial monoids studied in the literature, e.g., Catalan monoids and Kiselman monoids. This solves the Finite Basis Problem for stylic monoids.

We study the finite versus infinite nature of C$^{\\ast }$-algebras arising from étale groupoids. For an ample groupoid$G$, we relate infiniteness of the reduced C$^{\\ast }$-algebra$\\text{C}_{r}^{\\ast }(G)$to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid$S(G)$which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid characterizes the finite/infinite nature of the reduced groupoid C$^{\\ast }$-algebra of$G$in the sense that if$G$is ample, minimal, topologically principal, and$S(G)$is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for$\\text{C}_{r}^{\\ast }(G)$. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph$\\text{C}^{\\ast }$-algebras as well.

Let S be a numerical monoid, while a Pfin(S)-monoid S is a monoid generated by a finite number of finite non-empty subsets of S. That is, S is a non-cancellative commutative monoid obtained from the sumset of finite non-negative integer sets. This work provides an algorithm for computing the ideals associated with some Pfin(S)-monoids. These are the key to studying some factorization properties of Pfin(S)-monoids and some additive properties of sumsets. This approach links computational commutative algebra with additive number theory.