The concept of implicative n.p.o. (negatively partially ordered) ternary semigroups, which extends the concept of implicative n.p.o. semigroups introduced by M. W. Chan and K. P. Shum, Homomorphisms of implicative semigroups, Semigroup Forum 46 (1993) 7-15, was studied by K. Nakwan, P. Luangchaisri and T. Changphas, Implicative negatively partially ordered ternary semigroups, Eur. J. Pure Appl. Math. 17(4) (2024) 4180-4194. In this paper, we continue the study of these ternary semigroups and establish a connection between implicative homomorphisms and ternary semigroup homomorphisms of implicative n.p.o. ternary semigroups.

Let e be an idempotent of a semigroup. In this paper, we introduce the concept of left e-prime ideals and left e-domains of a semigroup. We investigate their algebraic properties. Moreover, we characterize a left e_{\rho_I}-domain by a left e-prime ideal I. Furthermore, we investigate left e-prime ideals on the cartesian product of two semigroups.

In this paper, we consider the concepts of uni-soft left and uni-soft right ideals, uni-soft quasi-ideals, and uni-soft bi-ideals within the context of ordered semigroups. We demonstrate that in ordered semigroups, both uni-soft right and uni-soft left ideals exhibit properties of uni-soft quasi-ideals. Similarly, uni-soft quasi-ideals possess characteristics of uni-soft bi-ideals. Furthermore, our analysis establishes that the definitions of uni-soft quasi-ideals and uni-soft bi-ideals align, indicating their equivalence within this specific class of semigroups. Additionally, we prove that in an ordered semigroup, uni-soft quasi-ideals can be understood simply as the unions of unisoft right and uni-soft left ideals. This elucidates the relationship between these concepts, shedding light on their fundamental role in the structure of ordered semigroups.

Suppose S is a F-semigroup with identity element e and zero element 0 such that e =6 0. For a proper subset A of S, the subset HA(S, F) C S is defined as follows: HA(S,F)= {hESfor all sES \ A, s E/ sFhFS}. That is, HA(S, F) consists of all elements h E S such that for every s E/ A, the element s does not belong to the set sFhFS. The following results are demonstrated: If A is a proper right ideal of S, then HA(S, F) is a semiprime ideal of S, and if A is a proper ideal of S, then HA(S, F) is a semiprime ideal of S containing A.

The aim of this paper is to present how implicative negatively partially ordered ternary semigroups can be induced from implicative negatively partially ordered semigroups.

In this paper, we review the concept of (m,n)-anti-ideals in semigroups. This notion extends the classical concept of anti-ideals and is closely related to the study of mutants in semigroups. It is shown that a semigroup possessing a left identity admits no left (m,m)-anti-ideals, and that no (m,n)-anti-ideals exist when m < n. These preliminary results serve as a foundation for the subsequent development of the theory in the context of ordered semigroups.

We study rough approximations in ternary groupoids via congruences and stable congruences. Upper and lower rough ternary subgroupoids and n-left ideals are introduced. We prove that every ternary subgroupoid forms an upper rough ternary subgroupoid (for any congruence) and a lower rough ternary subgroupoid (for any stable congruence). Likewise, each n-left ideal induces upper and lower rough n-left ideals.

We introduce and investigate the concept of (left, right, middle, two-sided) anti-ideals in ternary semigroups. Additionally, we establish a condition under which a ternary semigroup does not possess any anti-ideals.

Let $S_i$ be a semigroup for all $i \in \{1,2,\ldots,n\}$. Then the Cartesian product of $S_1, S_2, \ldots, S_n$ becomes a semigroup under componentwise multiplication. Let $(s_1,s_2,\ldots,s_n) \in S_1 \times S_2 \cdots \times S_n$. In this paper, we give necessary and sufficient condition when the Cartesian product of principal left ideals $L(s_1)\times L(s_2) \times \cdots \times L(s_n)$ is the principal left ideal $L((s_1,s_2,\ldots,s_n))$ and {the Cartesian} product of $\mathcal{L}$-classes$L_{s_1}\times L_{s_2} \times \cdots \times L_{s_n}$ is an $\mathcal{L}$-class $L_{(s_1,s_2,\ldots,s_n)}$ in a semigroup $S_1 \times S_2 \times \cdots \times S_n$.

Let $\tau_{n} = (n_i)_{i \in I}$ be a particular language (type) of algebras such that $n_i = n$ for all $i$ in $I$; $n$ is a positive integer. This paper aims to introduce \(n\)-ary alternating terms (alt-terms) of type \(\tau_n\), based on the alternating group \(Alt(n)\) of degree \(n\). We demonstrate that the set of all \(n\)-ary alternating terms of type \(\tau_n\) forms a Menger algebra of rank \(n\); such algebra is denoted by \({\mathcal W}^{Alt(n)}_{\tau_n}(\Omega_n)\). We prove that the algebra \({\mathcal W}^{Alt(n)}_{\tau_n}(\Omega_n)\) is free with respect to the variety \(V_{Menger}\) of Menger algebras of rank \(n\), and it is freely generated by the set \(\{\omega_{(i,\sigma)} : i \in I, \sigma \in Alt(n)\}\). We introduce alternating hypersubstitutions of type \(\tau_n\) and prove that the extension of an alternating hypersubstitution of type $\tau_n$ acts as an endomorphism on the algebra \({\mathcal W}^{Alt(n)}_{\tau_n}(\Omega_n)\). Furthermore, we have that the set of all alternating hypersubstitutions of type \(\tau_n\) forms a monoid, denoted by ${\mathcal Hyp}^{Alt(n)}(\tau_n)$. Finally, we establish that the set of all identities \(s \approx t\) of a variety \(V\) of type \(\tau_n\), where \(s\) and \(t\) are \(n\)-ary alternating terms of type \(\tau_n\), constitutes a congruence on the algebra \({\mathcal W}^{Alt(n)}_{\tau_n}(\Omega_n)\). According to the monoid ${\mathcal Hyp}^{Alt(n)}(\tau_n)$, we investigate alternating hyperidentities and alternating closed vareities.

Let Gamma be a nonempty set. A nonempty set A is called a Gamma-AG-groupoid if there is a function f from A x Gamma x A into A, customary denoted a gamma b for f (a, gamma, b), satisfying the identity (a gamma b)/3c = (c gamma b)/3a for any a, b, c E A and gamma, /3 E Gamma. For each gamma E Gamma, an operation on A associated to gamma is given by ab = a gamma b. Suppose further that A is finite, contains a left identity and a left zero a0. The objective of this paper is to provide sufficient conditions under which the set A \ {a0} is a commutative group under the operation on A determined by gamma for all gamma E Gamma.

In this paper, we study a special set in an implicative n.p.o.(negatively partially ordered) ternary semigroup, and prove that a filter can be represented by the union of such sets. Indeed, let $(T, [\,\,\,],\leq,[\,\,\,]^*)$ be an implicative n.p.o. ternary semigroup. For any $a, b\in T$, we define $$S(a,b):=\{c\in T \,:\, [aa[bbc]^*]^*=1\}.$$ We have the following:\begin{enumerate} \item [(1)] A non-empty subset $F$ of $T$ isa filter if and only if it satisfies the following conditions: \begin{enumerate} \item[(F3)] $1\in F$; \item[(F4)] for any $a, b,c \in T$, if $[abc]^*\in F$ and $a,b \in F$, then $c \in F$. \end{enumerate} \item [(2)] If $T$ is commutative and $F$ is a filter of $T$, then $$F=\displaystyle\bigcup_{a,b\in F} S(a,b).$$\end{enumerate}

Let $S$ be a semigroup and $x \in S$. The principal quasi-ideal of $S$ containing $x$ is denoted by $Q(x)$. An $\mathcal{H}$-class of $S$ containing $x$ is denoted by $H_x$. Let $S_1, S_2$ be semigroups. The direct product $S_1 \times S_2$ is defined as the Cartesian product of $S_1$ and $S_2$ equipped with the componentwise binary operation. Let $(a,b) \in S_1 \times S_2$. The direct product of $Q(a) \times Q(b)$ need not to be $Q((a,b))$. In this paper, we provide necessary and sufficient conditions when $Q(a) \times Q(b) = Q((a,b))$ and the conditions when $H_{(a,b)} = H_{a} \times H_{b}$.

The concept of Gamma-seminearrings is recognized as a generalization of seminearrings. In this paper, we study the notion of derivations on a Gamma-seminearring related to the notion of 3-prime strong ideals.

It is known that if S is a regular semigroup or a regular ordered semigroup, then the quasi-ideals and the bi-ideals of S coincide. The converse does not hold in general [4]. The purpose of this paper is to show that the results mentioned are true for hypersemigroups and ternary hypersemigroups.

In this paper, we study right weakly regular ternary semirings and fully prime right ternary semirings. Let T be a ternary semiring with absorbing zero and identity. We prove the following: (1) T is right weakly regular if and only if [AAA] = A for each right ideal A of T, and (2) T is a fully prime right ternary semiring if and only if T is right weakly regular and for ideals A, B and C of T one of the following assertions holds: A C B pi C; B C A pi C; C C A pi B.

An ordered semigroup is a semigroup (S, ·) together with a partial order ≤ on S such that x ≤ y implies z ·x ≤ z ·y and x ·z ≤ y ·z for all x,y,z in S .I f (S, ·, ≤S ) and (T,·, ≤T ) are two ordered semigroups, then the Cartesian product S × T is a semigroup under the coordinatewise multiplication. Define a partial order ≤ on S ×T by (s1,t1) ≤ (s2,t2 )i f and only if s1 ≤S s2 and t1 ≤T t2 for all (s1,t1), (s2,t2) ∈ S × T . Then S × T is an ordered semigroup. In this note, necessary and sufficient conditions of a subset of S × T to be a prime ideal will be presented.

In this paper, we focus on terms with fixed variables count, terms under which the total numbers of occurrences of variables in each position are equal. Moreover, we determine conditions for which the set of terms with fixed variables count is closed under the generalized superposition. Furthermore, we form the partial algebras of such terms satisfying certain axioms as weak identities.

In this paper, we introduce and examine the notion of implicative negatively partially ordered ternary semigroups, for short implicative n.p.o. ternary semigroup, which include an element that serves as both the greatest element and the multiplicative identity. We study the notion of implicative homomorphisms between these ternary semigroups, and have that any implicative homomorphism is a homomorphism. Let phi : T1 -> T2 be an implicative homomorphism from a commutative implicative n.p.o. ternary semigroup T1 onto T2. We construct a quotient commutative implicative n.p.o. ternary semigroup T1/rho Ker phi, where rho Ker phi is a congruence relation defined by Ker phi. We prove that there exists an implicative homomorphism psi such that psi degrees eta = phi, where eta is a canonical homomorphism from T1 onto T1/rho Ker phi.