A partial one-to-one mapping on a set X is a mapping whose domain is a subset of X. The set of partial one-one mappings on a set X is an inverse semigroup (: symmetric inverse semigroup on X ). An element x of an ordered semigroup (S, center dot, <=) is called an inverse element of alpha is an element of S if alpha <= x alpha x and x <= x alpha x. An inverse ordered semigroup is an ordered semigroup S for which every element of S possesses an inverse element and the inverse elements of any element of S are in the same connected component of the Hasse diagram of its order relation. In the paper, we give an analogous result to the Vagner-Preston Representation for inverse ordered semigroups by proving that for every inverse ordered semigroup S there exists an R-<=- monoantimorphism from S to the symmetric inverse semigroup on S.

An element e of an ordered semigroup ( S,· ,≤) is called idempotent (resp. generalised idempotent) if e≤e^2 (resp. ( e,e^2) ∈_≤ where _≤ is the smallest congruence on S containing the relation ≤∪≤^-1 ). The set of all idempotents (resp. generalised idempotents) of S is denoted by E( S ) (resp. E^G( S ) ). S is called orthodox if (i) the set E( S ) is non empty and (ii) ef∈E^G( S ) for every e,f∈ E( S ) . An element x in S is an inverse (resp. generalised inverse) of an element a of S if a≤ axa and x≤ xax (resp. ( a,axa ) ,( x,xax ) ∈_≤ ). We study the notions of generalised inverse and generalised idempotent element and we show that, in an orthodox ordered semigroup, if we know a single generalised inverse of an element a, then we know the set of all generalised inverses of a. We also study the structure of orthodox ordered semigroups giving basic properties of orthodox ordered semigroups and equivalent conditions (based on inverse, generalised inverse, idempotent, generalised idempotent elements) according to which an ordered semigroup is orthodox.

An element x of an ordered semigroup ( S, center dot, <= ) is called an inverse element of alpha is an element of S if alpha <= x alpha x and x <= x alpha x. An inverse ordered semigroup is an ordered semigroup S for which every element of S possesses an inverse element and the inverse elements of any element of S are in the same connected component of the Hasse diagram of its order relation. We present properties satisfied by inverse ordered semigroups and characterize inverse ordered semigroups based on the notion of regularity of ordered semigroups, their set of idempotents and Green's relations. Also we prove that the inversibility of ordered semigroups is preserved by homomorphisms.

A transformation of an ordered set M is an isotone mapping of M into M. By a representation of an ordered semigroup S by transformations of an ordered set M we mean a homomorphism of S into the set of transformations of M, i.e. (since the set of transformations of M is an ordered semigroup) an isotone mapping from S into the set of transformations of M preserving the operations. We prove that this type of representation leads to an “action” of S on M and so we introduce the notion of a left operand of M over S. Also we introduce the notions of a left operator pseudoorder on a left operand over S and a left operator homomorphism between left operands over S. We show that the concept of left operator pseudoorders on left operands over S plays an important role in the study of left operator homomorphisms of left operands over S. In the case of right operands over S dually definitions and results hold.

A type F of ordered semigroups is a class of ordered semigroups such that (i) if S belongs to F and S  is isomorphic to S, then S  belongs to F , and (ii) any one-element (ordered) semigroup belongs to F . Given a type F of ordered semigroups and an ordered semigroup S, the radS  F is the intersection of all pseudoorders of S having type F (a pseudoorder σ on S has type F if the quotient semigroup of S by the congruence 1 :       has also type F - we consider the quotient semigroup as an ordered semigroup under the induced order relation by σ). The derived type  F of F is the class of all ordered semigroups S such that radS  F is the order relation of S. An F maximal homomorphic image of an ordered semigroup of an ordered semigroup S is an ordered semigroup S  in F for which there exists a homomorphism η of S onto S  with the factorization property: if υ is a homomorphism of S onto an ordered semigroup of type F , then there exists a homomorphism θ of S  onto T such that      . We give sufficient and necessary condition under which an ordered semigroup admits an F maximal homomorphic image. We show that every ordered semigroup has an  F maximal homomorphic image and finally for a type F of ordered semigroups we prove thatevery ordered semigroup has an F maximal homomorphic image if and only if

Given an indexed family $\left\{ \left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right),i\in I \right\}$ of disjoint ordered semigroups, we construct an ordered semigroup having $\left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right)$, $i\in I$ as subsemigroups (with respect to the operation and order relation of each $\left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right)$, $i\in I$). This ordered semigroup is the free ordered product ${{\underset{i\in I}{\mathop{\Pi }}\,}^{*}}{{S}_{i}}$ of the family $\left\{ {{S}_{i}},i\in I \right\}$ and we give the crucial property which essentially characterizes the free products. Next we study the same problem in the case that the family $\left\{ \left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right),i\in I \right\}$ of ordered semigroups has as intersection the ordered semigroup $\left( U,{{\cdot }_{U}},{{\le }_{U}} \right)$ which is a subsemigroup of $\left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right)$ for every $i\in I$ (with respect to the operation and order relation of each $\left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right)$, $i\in I$). To do this, we first consider the ordered semigroup amalgam $\mathfrak{A}=\left[ \left\{ \left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right),i\in I \right\};\left( U,{{\cdot }_{U}},{{\le }_{U}} \right);\left\{ {{\varphi }_{i}}:U\to {{S}_{i}},i\in I \right\} \right]$ (where $\left\{ {{\varphi }_{i}}:U\to {{S}_{i}},i\in I \right\}$ is a family of monomorphisms) and then we construct the free ordered product $\underset{i\in I\text{ }}{\mathop{\Pi _{U}^{*}}}\,{{S}_{i}}$ of the ordered semigroup amalgam $\mathfrak{A}$ considering the ordered quotient of the free ordered product ${{\underset{i\in I}{\mathop{\Pi }}\,}^{*}}{{S}_{i}}$ by an appropriate pseudoorder of ${{\underset{i\in I}{\mathop{\Pi }}\,}^{*}}{{S}_{i}}$ through which for each $i,j\in I$ and for each $u\in U$, ${{\varphi }_{i}}\left( u \right)\in {{S}_{i}}$ is identified (by means of monomorphisms) with ${{\varphi }_{j}}\left( u \right)\in {{S}_{j}}$. We give a sufficient and necessary condition so that an ordered semigroup amalgam is embedded in an ordered semigroup. At the end of the paper, we introduce the notion of ordered dominions. An element $d$ of an ordered semigroup $S$ is dominated by a subsemigroup $U$ of $S$ if for all ordered semigroups $\left( T ,\cdot ,\le \right)$ and for all homomorphisms $\beta ,\gamma :S\to T$ such that $\beta \left( u \right)=\gamma \left( u \right)$ for each $u\in U$, we have $\left[ \beta \left( d \right) \right)_{\le }^{T}\cap \left[ \gamma \left( d \right) \right)_{\le }^{T}\ne \varnothing $. In the last Theorem of the paper, we give an expression of the set of elements of $S$ dominated by $U$ based on ordered semigroup amalgams.

Given an indexed family {(S-i, center dot(i),<= (i)), i is an element of I} of disjoint ordered semigroups, we construct an ordered semigroup having (S-i, center dot(i),<=(i)), i is an element of I as subsemigroups (with respect to the operation and order relation of each (S-i, center dot(i),<= (i)), i is an element of I. This ordered semigroup is the free ordered product ||(i is an element of I) * S-i of the family {S-i, i is an element of I} and we give the crucial property which essentially characterizes the free products. Next we study the same problem in the case that the family {(S-i, center dot(i),<= (i)), i is an element of I} of ordered semigroups has as intersection the ordered semigroup (U, center dot(U),<=(U)) which is a subsemigroup of (S-i, center dot(i),<= (i)) for every i is an element of I (with respect to the operation and order relation of each (S-i, center dot(i),<= (i)), i is an element of I). To do this, we first consider the ordered semigroup amalgam U = [{(S-i, center dot(i),<= (i)), i is an element of I}; (U, center dot(U),<=(U)); {phi(i) : U -> S-i, i is an element of I}] (where {phi(i) : U -> S-i, i is an element of I} is a family of monomorphisms) and then we construct the free ordered product Pi(i is an element of IU)* S-i of the ordered semigroup amalgam U considering the ordered quotient of the free ordered product Pi(i is an element of I)* S-i by an appropriate pseudoorder of Pi(i is an element of IU)* S-i through which for each i, j is an element of I and for each u is an element of U, phi(i) (u) is an element of S-i is identified (by means of monomorphisms) with phi(j) (u) is an element of S-j. We give a sufficient and necessary condition so that an ordered semigroup amalgam is embedded in an ordered semigroup. At the end of the paper, we introduce the notion of ordered dominions. An element d of an ordered semigroup S is dominated by a subsemigroup U of S if for all ordered semigroups (T, center dot,<=) and for all homomorphisms beta,gamma : S -> T such that beta(u) = gamma(u) for each u is an element of U, we have [beta (d))(T) <= boolean AND [gamma (d))T-<= not equal(sic). In the last Theorem of the paper, we give an expression of the set of elements of S dominated by U based on ordered semigroup amalgams.

We introduce the concepts of unitary, almost unitary and strongly almost unitary subset of an ordered semigroup.For the notions of almost unitary and strongly almost unitary subset of an ordered semigroup, we use the notion of translational hull of an ordered semigroup.If ( ) , , S ⋅ ≤ is an ordered semigroup having an element e such that 2 e e ≤ and U is a nonempty subset of S such that u eu ≤ , u ue ≤ for all u U ∈ , we show that U is almost unitary in S if and only if U is unitary in ( ] ( )

This note defines the left magnifying and the strongly left magnifying elements in an ordered groupoid and discusses their properties. It is shown that in an ordered semigroup, every left magnifying element is of infinite order. The concept of factorizable ordered semigroups has been also introduced and, using the infinite order property, it is shown that every ordered semigroup having a strongly left magnifying element is factorizable.

A fuzzy subset f of an ordered groupoid (or groupoid) S is called fuzzy semiprime if f(x) >= f (x(2)) for every x is an element of S; it is called fuzzy prime if f(x) <= min{f (x), f (y)} for every x, y is an element of S (Definition 1). Following the terminology of semiprime subsets of ordered groupoids (or groupoids) and the terminology of ideal elements of poe-groupoids (: ordered groupoids possessing a greatest element), a fuzzy subset f of an ordered groupoid (or groupoid) should be called fuzzy semiprime if for every fuzzy subset g of S such that g(2) := g o g <= f, we have g <= f; it should be called prime if for any fuzzy subsets h, g of S such that h o g <= f we have h <= f of g <= f (Definition 2). And this is because if S is a groupoid or an ordered groupoid, then the set of all fuzzy subsets of S is a poe-groupoid. What is the relation between these two definitions? that is between the Definition 1 (the usual definition we always use) and the Definition 2 given in this paper? The present paper gives the related answer.

For an intra-regular or a left regular and left duo ordered $\Gamma$-semigroup $M$, we describe the principal filter of $M$ which plays an essential role in the structure of this type of $po$-$\Gamma$-semigroups. We also prove that an ordered $\Gamma$-semigroup $M$ is intra-regular if and only if the ideals of $M$ are semiprime and it is left (right) regular and left (right) duo if and only if the left (right) ideals of $M$ are semiprime.

For an intra-regular or a left regular and left duo ordered Γ-semigroup M, we describe the principal filter of M which plays an essential role in the structure of this type of po-Γ-semigroups. We also prove that an ordered Γ-semigroup M is intra-regular if and only if the ideals of M are semiprime and it is left (right) regular and left (right) duo if and only if the left (right) ideals of M are semiprime.

We characterize the set of regular elements of an ordered semigroup S which is a left (right) ideal, an ideal, a bi-ideal or a quasi-ideal of S using the left (resp. right) ideal L-E(S) = boolean OR(e is an element of E(S)) (Se](resp. R-E(S) = boolean OR(e is an element of E(S)) (eS]), the ideal I-E(S) = boolean OR(e is an element of E(S)) (SeS], the bi-ideal B-E(S) = boolean OR(e,f is an element of E(S)) (eSf], and the quasi-ideal L-E(S) boolean AND R-E(S) of S, where E(S) is the set of elements of S for which e <= e(2). Illustrative example is given. (C) 2015 Mathematical Institute Slovak Academy of Sciences

We prove that if an ordered semigroup is a nil extension of a left strongly simple ordered semigroup, then it is left strongly archimedean, but, in contrast to the unordered case, the converse does not hold in general. However, a left strongly archimedean ordered semigroup is a nil extension of a simple ordered semigroup.

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.

A fuzzy subset f of an ordered semigroup (or semigroup) S is called fuzzy semiprime if f(x)≥ f(x^2) for every x∈ S (Definition 1). Following the terminology of semiprime subsets of ordered semigroups (semigroups), the terminology of ideal elements of poe-semigroups (: ordered semigroups possessing a greatest element), and the terminology of ordered semigroups, in general, a fuzzy subset f of an ordered semigroups (semigroup) should be called fuzzy semiprime if for every fuzzy subset g of S such that g^2:=g∘ g≼ f, we have g≼ f (Definition 2). And this is because if S is a semigroup or ordered semigroup, then the set of all fuzzy subsets of S is a semigroup (ordered semigroup) as well. What is the relation between these two definitions? that is between the usual definition (Definition 1) we always use and the definition we give in the present paper (Definition 2) saying that that definition should actually be the correct one? The present paper gives the related answer.

We prove that a Γ-semigroup M is left (resp. right) regular if and only if it is decomposable into left (resp. right) simple subsemigroups of M .

Concerning the paper in the title by K. Hila and E. Pisha in Commun. Korean Math. Soc. Volume 26, Issue 3 (2011), 373--384, we give our results and make the main corrections.

Comments on the paper in the title published in Communications Korean Mathematical Society. We give our results and make the main corrections. Mathematics Subject Classification: 06F99 (06F05)

According to the paper in [3], the kernel of an ordered semigroup S is a completely regular subsemigroup of S. In this note we show that the kernel of an ordered semigroup S is not a completely regular subsemigroup of S, in general.