Semigroup Forum最新文献

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 ({mathscr {P}}_{n+1}) of uniform block permutations on the set ({1,2,ldots , n+1}). Some classical and known presentations of ({mathscr {P}}_{n+1}), including FitzGerald’s presentation and Everitt and Fountain’s presentation, are recovered.

We consider the stabilization of two weakly dissipative wave equations with variable coefficients, localized Kelvin–Voigt damping, mixed boundary condition and time delay. The well posedness is obtained by the semigroup method. Using a unique continuation result, we show that the system is strongly stable. Then, we show that the system is not exponentially stable. Finally, using a frequency domain approach combined with multiplier method, we establish a polynomial energy decay rate for the undelayed system. Then, we prove that the system with delay has the same decay rate.

In 1981, Gerhard Gierz and Jimmie Lawson gave a necessary condition for a complete lattice to be quasicontinuous using an equation. However, whether quasicontinuous lattices can be characterized by equations has remained unknown. In this paper, we give an equational characterization of quasicontinuous complete semilattices. Furthermore, we investigate congruences of quasicontinuous complete semilattices. We derive the first isomorphism theorem for quasicontinuous complete semilattices in the context of C-congruences. As a consequence, we show that the category of all continuous complete semilattices with maps preserving directed sups and nonempty infs is a reflective full subcategory of quasicontinuous complete semilattices.

We consider a (C_0)-semigroup on a Banach space such that the resolvent is uniformly bounded on the right half-plane. In this paper we provide a condition on the resolvent which is sufficient and necessary for the uniform exponential stability of such a semigroup. As a consequence, we give an alternative proof of Gearhart’s theorem (Trans. Amer. Math. Soc. 236, 385–394 (1978)). The approach lies on a complex inversion formula and tempered distributions.

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.

Følner density is a very natural notion of density which is defined on any semigroup satisfying the Strong Følner Condition (SFC). (These include all commutative semigroups and all left cancellative left amenable semigroups.) Piecewise syndetic and thick are notions of largeness arising from topological dynamics. It has been known that if S satisfies SFC and is either left cancellative or satisfies a weak right cancellation requirement, then every thick subset has density 1. We show here that in any semigroup S satisfying SFC a subset of S is thick if and only if it has density 1. As a consequence, every piecewise syndetic set has positive density.

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.

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.

It is shown that the Hecke–Kiselman monoid ({text {HK}}_{Theta }) associated to a finite oriented graph (Theta ) satisfies a semigroup identity if and only if ({text {HK}}_{Theta }) does not have free noncommutative subsemigroups. It follows that this happens exactly when the semigroup algebra (K[{text {HK}}_{Theta }]) over a field K satisfies a polynomial identity. The latter is equivalent to a condition expressed in terms of the graph (Theta ). The proof allows to derive concrete identities satisfied by such monoids ({text {HK}}_{Theta }).

The notions of a CR set is intimately related with the generalized van der Waerden’s theorem. We prove that the product of two CR sets is again a CR set. This answers Question 4.2 from Hindman et al. (Semigroup Forum 107:127–143, 2023).