Commutative semigroups whose endomorphisms are power functions

Semigroup Forum - 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...     

Commutative semigroups all of whose subsemigroups are densely embedded

Acta Mathematica Hungarica -     

Commutative semigroups having greatest regular images

Let S be a commutative semigroup. S has a greatest regular image if and only if each of its Archimedean components contains an idempotent. S has a greatest group-with-zero image if and only if S has precisely two Archimedean components and the upper component contains an idempotent. The existence and structure of these images and of greatest group images is related to tensor products.     

Commutative Noetherian semigroups are finitely generated

We provide a short proof that a commutative semigroup is finitely generated if its lattice of congruences is Noetherian.     

Commutative semigroups which are embeddable into epigroups

Communicated by L. N. Shevrin     

Commutative Rings whose Proper Ideals are Serial

A result of Nakayama and Skornyakov states that a ring R is an Artinian serial ring if and only if every R-module is serial. This motivated us to study commutative rings for which every proper ideal is serial. In this paper, we determine completely the structure of commutative rings R of which every proper ideal is serial. It is shown that every proper ideal of R is serial, if and only if, either R is a serial ring, or R is a local ring with maximal ideal \({\mathcal {M}}\) such that there exist a uniserial module U and a semisimple module T with \({\mathcal {M}}=U\oplus T\). Moreover, in the latter case, every proper ideal of R is isomorphic to \(U^{\prime }\oplus T^{\prime }\), for some \(U^{\prime }\leq U\) and \(T^{\prime }\leq T\). Furthermore, it is shown that every proper ideal of a commutative Noetherian ring R is serial, if and only if, either R is a finite direct product of discrete valuation domains and local Artinian principal ideal rings, or R is a local ring with maximal ideal \({\mathcal {M}}\) containing a set of elements {w ₁,…,w _n } such that \({\mathcal {M}}=\bigoplus _{i=1}^{n} Rw_{i}\) with at most one non-simple summand. Moreover, another equivalent condition states that: there exists an integer n ≥ 1 such that every proper ideal of R is a direct sum of at most n uniserial R-modules. Finally, we discuss some examples to illustrate our results.     

Polycyclic groups with modular finite homomorphic images

A group G is said to be a modular group if it has modular subgroup lattice. We will prove in this paper that a polycyclic group G is modular if and only if all its finite homomorphic images are modular groups. Similar results will also be obtained for other conditions of modular type.     

Conjugacy classes in finite permutation groups via homomorphic images

The lifting of results from factor groups to the full group is a standard technique for solvable groups. This paper shows how to utilize this approach in the case of non-solvable normal subgroups to compute the conjugacy classes of a finite group.

     

Bisimple inverse semigroups whose idempotents are an interval

Bisimple locally compact topological inverse semigroups whose maximal subgroups are compact and whose set of idempotents is a real interval are studied in this paper. The structure of such semigroups is shown to be entirely determined by their maximal subgroups in case the semigroups come from a certain subclass which includes the connected ones. The factor semigroups moduloH are completely determined.     

Weierstrass semigroups whose minimum positive integers are even

We consider three subsets of the set of 2n-semigroups, where for a positive integer n a 2n-semigroup means a numerical semigroup whose minimum positive integer is 2n. These three subsets are obtained by the Weierstrass semigroups of total ramification points on a cyclic covering of the projective line, the Weierstrass semigroups of ramification points on a double covering of a non-singular curve and the Weierstrass semigroups of points on a non-singular curve. We show that the three subsets are different for n ≧ 3. Partially supported by Grant-in-Aid for Scientific Research (17540046), Japan Society for the Promotion of Science. Received: 19 June 2006     

真理想为Archimedean子半群的偏序半群

根据真理想情况给出了偏序半群的一种分类，研究了真理想为Archimedean子半群的偏序半群的特征．