Unitary posets and amalgamations of pomonoids

In 1927, Schreier proved that amalgams of groups are always embeddable in the category of groups. However, this is not true in the category of semigroups, as shown by Kimura. Subsequently, Howie initiated the study of semigroup amalgams by investigating when the embeddablity happens, and found that semigroup amalgams can be embeddable if the core of the amalgam is almost unitary [18]. Later, Hall proved that inverse semigroups are amalgamation bases in the category of inverse semigroups [14], and Renshaw introduced a homological structure in order to describe the amalgamated free product [32]. By using this structure, Renshaw proved that a semigroup U is an amalgmation base if, and only if, U has the extension property in every containing semi-group. Renshaw's result, which shows that a semigroup amalgam is embeddable if, and only if, it is embeddable as a monoid, allow us to focus on monoid amalgams. The subject of pomonoid amalgams was first studied by Fakhuruddin in 1986 but he only considered the commutative case [10]. Little work has been done in this category and recently Bulman-Fleming and Nasir revisited this area (see [7], [6], and [29]). They modified Fakhuruddin's definition of pomonoid amalgams, where they proved that a pomonoid amalgam that has the postrong representation extension property is strongly poembeddable [7]. They also proved that pogroups are strong poamalgamation bases in the category of pomonoids. Nasir [29] found that absolutely poatness pomonoids are strong poamalgamation bases in the category of commutative pomonoids. However, several questions remain unanswered in this area, and this research continues to study pomonoid amalgams by exploring when poembeddability can happen. It also aims to generalise some of the results in monoid amalgams. In addition, a number of subjects related to pomonoid amalgams have been considered, for example dominions and subpomonoid amalgams. New questions about the class of amalgamation bases have emerged recently and we briefly consider some of these

Vertex-edge perfect Roman domination number

A vertex-edge perfect Roman dominating function on a graph $G = (V, E)$ (denoted by ve-PRDF) is a function $f:V\left(G\right)\longrightarrow\{0, 1, 2\}$ such that for every edge $uv\in E$, $\max\{f(u), f(v)\}\neq0$, or $u$ is adjacent to exactly one neighbor $w$ such that $f(w) = 2$, or $v$ is adjacent to exactly one neighbor $w$ such that $f(w) = 2$. The weight of a ve-PRDF on $G$ is the sum $w(f) = \sum_{v\in V}f(v)$. The vertex-edge perfect Roman domination number of $G$ (denoted by $\gamma_{veR}^{p}(G)$) is the minimum weight of a ve-PRDF on $G$. In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree $T$, we give upper and lower bounds for $\gamma_{veR}^{p}(T)$ in terms of the order $n$, $l$ leaves and $s$ support vertices. Lastly, we determine $\gamma_{veR}^{p}(G)$ for Petersen, cycle and Flower snark graphs

On the Monoid of Unital Endomorphisms of a Boolean Ring

Let X be a nonempty set and P(X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End1P(X)(P(X)) of unital ring endomorphisms of the Boolean ring P(X) and the automorphism group Aut(P(X)) when X is finite. Among other facts, it is shown that if X has cardinality n≥1, then End1P(X)(P(X))≅Tnop, where Tn is the full transformation monoid on the set X and Aut(P(X))≅Sn

On pomonoid of partial transformations of a poset

The main objective of this article is to study the ordered partial transformations PO(X){\mathcal{PO}}\left(X) of a poset XX. The findings show that the set of all partial transformations of a poset with a pointwise order is not necessarily a pomonoid. Some conditions are implemented to guarantee that PO(X){\mathcal{PO}}\left(X) is a pomonoid and this pomonoid is denoted by PO↑(X){{\mathcal{PO}}}^{\uparrow }\left(X). Moreover, we determine the necessary conditions in order that the partial order-embedding transformations define the ordered version of the symmetric inverse monoid. The findings show that this set is an inverse pomonoid and we will denote it by ℐPO↑(X){{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X). In case the order on the poset XX is total, we explore some properties of PO↑(X){{\mathcal{PO}}}^{\uparrow }\left(X) and ℐPO↑(X){{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X), including regressive, unitary, and reversible

On the Monoid of Unital Endomorphisms of a Boolean Ring

Let X be a nonempty set and P(X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End1P(X)(P(X)) of unital ring endomorphisms of the Boolean ring P(X) and the automorphism group Aut(P(X)) when X is finite. Among other facts, it is shown that if X has cardinality n≥1, then End1P(X)(P(X))≅Tnop, where Tn is the full transformation monoid on the set X and Aut(P(X))≅Sn

On subamalgams of partially ordered monoids

The study of pomonoid amalgams was initiated by Fakhuruddin in the 1980s and subsequently extended by Bulman-Fleming, Sohail and the authors in the 2000s. We further investigate pomonoids amalgams and in particular we consider the concept of subpomonoid amalgams possessing a suitable ordered version of the unitary property. If [U; T 1, T 2] is an amalgam of subpomonoids of the amalgam [U; S 1, S 2] we consider the question of whether the free product of the pomonoid amalgam [U; T 1, T 2] is poembeddable in the free product of the pomonoid amalgam [U; S 1, S 2] , giving a sufficient condition in terms of strongly pounitary subpomonoids. </p

On free products and amalgams of pomonoids

The study of amalgamation in the category of partially ordered monoids was initiated by Fakhuruddin in the 1980s. In 1986 he proved that, in the category of commutative pomonoids, every absolutely flat commutative pomonoid is a weak amalgmation base and every commutative pogroup is a strong amalgamation base. Some twenty years later, Bulman-Fleming and Sohail in 2011 extended this work to what they referred to as pomonoid amalgams. In particular, they proved that pogroups are poamalgmation bases in the category of pomonoids. Sohail, also in 2011, proved that absolutely poflat commutative pomonoids are poamalgmation bases in the category of commutative pomonoids. In the present article, we extend the work on pomonoid amalgams by generalizing the work of Renshaw on amalgams of monoids and extension properties of acts over monoids