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.

On the endomorphism monoids of Clifford semigroups

In this paper, we study properties of the endomorphism monoids of strong semilattices of groups. In Sec. 2, several properties for endomorphism monoids of finite semilattices are investigated. In Sec. 3, we collect some results on endomorphism monoids of strong semilattices of groups, i.e. Clifford semigroups.

THE REDUCTS OF THE HOMOGENEOUS BINARY BRANCHING C-RELATION

Let ($\rm L$;C) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of ($\rm L$;C), i.e., the structures with domain $\rm L$ that are first-order definable in ($\rm L$;C). We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinability, thus confirming a conjecture of Simon Thomas for the special case of ($\rm L$;C). We also study the endomorphism monoids of such reducts and show that they fall into four categories.

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.