For a numerical semigroup ring K[H] we study the trace of its canonical ideal. The colength of this ideal is called the residue of H. This invariant measures how far is H from being symmetric, i.e. how far is K[H] from being a Gorenstein ring. We remark that the canonical trace ideal contains the conductor ideal, and we study bounds for the residue. For 3-generated numerical semigroups we give explicit formulas for the canonical trace ideal and the residue of H. Thus, in this setting we can classify those whose residue is at most one (the nearly Gorenstein ones), and we show the eventual periodic behaviour of the residue in a shifted family.

In this paper we present and study the ideal duplication, a new construction within the class of the relative ideals of a numerical semigroup S, that, under specific assumptions, produces a relative ideal of the numerical duplication Sbowtie ^b E. We prove that every relative ideal of the numerical duplication can be uniquely written as the ideal duplication of two relative ideals of S; this allows us to better understand how the basic operations of the class of the relative ideals of Sbowtie ^b E work. In particular, we characterize the ideals E such that Sbowtie ^b E is nearly Gorenstein.

We define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. We consider the monoid generated by such functions. We call special idempotent any idempotent element of this monoid. They are interval retracts. Some of them realize a kind of parabolic map and are called special projections. We prove that, in Eulerian posets, the image of a special projection, and its complement, are graded induced subposets. In a finite Coxeter group, all projections on right and left parabolic quotients are special projections, and some projections on double quotients too. We extend our results to special partial matchings.

The irreducibility of pseudovarieties of semigroups is studied using many different approaches including a notion of Kovacs–Newman semigroups. In this note, we answer a natural question, whether Kovacs–Newman semigroups can be used in the study of the irreducibility of pseudovarieties of ordered semigroups.

For any commutative semigroup S and positive integer m the power function f: S rightarrow S defined by f(x) = x^m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

In this paper, two main results on the uniform stability of a resolvent family $$\{R_{h}(t)\}_{t\ge 0}$$ , depending on a parameter h are presented. First, we discuss a GGP type theorem on the resolvent family $$\{R_{h}(t)\}_{t\ge 0}$$ and give some sufficient conditions on the uniform stability of $$\{R_{h}(t)\}_{t\ge 0}$$ . Then we prove that under some suitable conditions, the weak $$L^{p}$$ -stability of $$\{R_{h}(t)\}_{t\ge 0}$$ implies its uniform stability. Our results both essentially generalize previous work on the uniform stability of a family of $$C_{0}$$ -semigroups $$\{T_{h}(t)\}_{t\ge 0}$$ and a resolvent family $$\{R(t)\}_{t\ge 0}$$ , without depending on the parameter h. Examples are also given to illustrate our results.

It has been known for a long time that every finite orthodox completely regular semigroup has a finite basis of identities, and that every finite central completely simple semigroup has a finite basis of identities. In the present paper, a common generalization of these two facts is established. It is shown that every finite central locally orthodox completely regular semigroup has a finite basis of identities. The proof of this latter fact which is presented in this paper employs significantly the celebrated theorem of Libor Polak on the structure of the lattice of all varieties of completely regular semigroups.

A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup $$\mathscr {FI}_{n}$$ of all order-preserving partial injections on an n-element fence. In particular, we provide a minimal generating set for $$\mathscr {FI}_{n}$$ . In the present paper, n is odd since this problem for even n was already solved by I. Dimitrova and J. Koppitz.

Two semigroups are called Morita equivalent if the categories of firm right acts over them are equivalent. We prove that every semigroup is Morita equivalent to its subsemigroup consisting of all products of n factors. Using this we show that a finite semigroup is Morita equivalent to its largest factorisable subsemigroup. It follows that two finite semigroups are Morita equivalent if and only if their Cauchy completions are equivalent categories. Since these categories are finite, the problem of Morita equivalence of finite semigroups is decidable.