On Closed Quasi Principally Injective Acts over Monoids

The concept of closed quasi principally injective acts over monoids is introduced ,which signifies a generalization for the quasi principally injective as well as for the closed quasi injective acts. Characterization of this concept is intended to show the behavior of a closed quasi principally injective property. At the same time, some properties of closed quasi principally injective acts are examined in terms of their endomorphism monoid. Also, the characterization of a closed self-principally injective monoid is given in terms of its annihilator. The relationship between the following concepts is also studied; closed quasi principally injective acts over monoids, Hopfian, co Hopfian, and directly finite property. Ultimately, based on the results obtained, the conditions on subacts to inherit a closed quasi principally injective property were shown. Part of this paper was dedicated to studying the relationship between the classes of closed quasi principally injective acts with some generalizations of injectivity. Conclusions and future remarks of this work are given.

Endomorphisms of Clifford semigroups with injective structure homomorphisms

In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective structure homomorphisms, where the semilattice has a least element. We describe such Clifford semigroups having a regular endomorphism monoid. If the endomorphism monoid on the Clifford semigroup is completely regular then the corresponding semilattice has at most two elements. We characterize all Clifford semigroups Gα∪Gβ (α>β) with an injective structure homomorphism, where Gα has no proper subgroup, such that the endomorphism monoid is completely regular. In particular, we consider the case that the structure homomorphism is bijective.

On Endomorphism Monoids of Finite Kleene Algebras

An endomorphism monoid of an algebra [Formula: see text] is said to be a band if every endomorphism on [Formula: see text] is an idempotent, and it is said to be a demi-band if every non-injective endomorphism on [Formula: see text] is an idempotent. We precisely determine finite Kleene algebras whose endomorphism monoids are demi-bands and bands via Priestley duality.

A note on subgroups of automorphism groups of full shifts

We discuss the set of subgroups of the automorphism group of a full shift and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case and free products in the group case. We also show that the automorphism group of a full shift embeds in that of an uncountable sofic shift. Some undecidability results are obtained as corollaries.

Unicyclic graphs with regular endomorphism monoids

The motivation of this paper comes from an open question: which graphs have regular endomorphism monoids? In this paper, we give a definitely answer for unicyclic graphs, proving that a unicyclic graph [Formula: see text] is End-regular if and only if, either [Formula: see text] is an even cycle with 4, 6 or 8 vertices, or [Formula: see text] contains an odd cycle [Formula: see text] such that the distance of any vertex to [Formula: see text] is at most 1, i.e., [Formula: see text]. The join of two unicyclic graphs with a regular endomorphism monoid is explicitly described.