Home

Research Interests

I am working on ideal theory in ordered semigroups. Let me provide some brief concepts of ideals in ordered semigroups as follows. Let $\mathbb{N}$ be the set of all natural numbers, and $\mathbb{N}_0$ the set of all nonnegative integers. That is, $\mathbb{N}_0 = \mathbb{N} \cup \{0\}$. For any $n \in \mathbb{N}$, we denote the set $\{1,\ldots,n\}$ by $[n]$.

An ordered semigroup is an algebraic system $\langle S; \cdot, \leq \rangle$ consisting of a nonempty set $S$, an associative binary operation $\cdot$ on $S$, and a partial order $\leq$ defined on $S$ such that the partial order is compatible with the binary operation $\cdot$.

If $\langle S; \cdot, \leq \rangle$ is an ordered semigroup, we denote the product of elements in $S$ by avoding the symbol $\cdot$. Since the operation $\cdot$ is associative, we can skip writing the parentheses. For an element $a$ of $S$ and any $n \in \mathbb{N}$, we write the $n$-product of $a$ by $a^n$. Moreover, we set $a^0b = b = ba^0$ for all $a,b \in S$. For the convenience, we denote an ordered semigroup $\langle S; \cdot, \leq \rangle$ by $\mathbf{S}$ the boldface of its underlying set.

Let $\mathbf{S}$ be an ordered semigroup. The operation $\cdot$ defined on $S$ can be extended to $\mathcal{P}(S)$ the set of all subsets of $S$, defined by $AB := \{ ab : a \in A \text{ and } b \in B \}$. We note here that $AB = \emptyset$ if $A$ or $B$ is empty. For a nonempty subset $A$ of $S$ and $n \in \mathbb{N}$, we denote the $n$-product of $A$ by $A^n$. We set $A^0B = B = BA^0$ for any nonempty subsets $A$ and $B$ of $S$. Moreover, the partial order $\leq$ induces a subset $(A]$ of $S$ defined by $(A] := \{ x \in S : x \leq a \text{ for some } a \in A \}$ for all $A \subseteq S$.

Now, let me recall some of the important substructures of ordered semigroups. Let $\mathbf{S}$ be an ordered semigroup. A nonempty subset $A$ of $S$ is said to be a subsemigroup of $\mathbf{S}$ if $AA \subseteq A$. We call a subsemigroup $A$ of an ordered semigroup $\mathbf{S}$ satisfying $(A] \subseteq A$:

In certain literatures, the subsemigroup property of bi-ideals and interior ideals is not necessary. However, in this note such a property is required.

The above ideals are classical substructures used to investigate several algebraic properties of ordered semigroups. Nevertheless, these notions are extended into the notions of $(m,n)$-ideals and $n$-interior ideals as follows.

Let $m, n \in \mathbb{N}$, not all zero. A subsemigroup $A$ of an ordered semigroup $\mathbf{S}$ satisfying $(A] \subseteq A$ is said to be:

We observe that a left ideal, a right ideal, a bi-ideal, and an interior ideal is a $(0,1)$-ideal, a $(1,0)$-ideal, a $(1,1)$-ideal, and a $1$-interior ideal. Therefore, the notions of $(m,n)$-ideals and $n$-interior ideals are generalizations of certain ideals.

However, the notions of $(m,n)$-ideals and $n$-interior ideals in ordered semigroups can be defined in a more abstract way. It was started in semigroup theory, in 1989, Miccoli and Pondělíček introduced the concept of $\alpha$-ideals in semigroups. In this note let we recall the notion of $\alpha$-ideals in the context of ordered semigroups. To this end, we need the concept of full words.

Let $X$ be an alphabet. We denote the free monoid and the free semigroup over $X$ by $X^*$ and $X^+$, respectively. The neutral element of $X^*$ is called the empty word, and it is denoted by $\varepsilon$. That is, $X^+ = X^* \smallsetminus \{ \varepsilon \}$. For $\alpha \in X^*$, the length of $\alpha$ is denoted by $\vert \alpha \vert$. We note that $\vert \alpha \vert = 0$ if $\alpha = \varepsilon$. If $X = \{0,1\}$, then we write $X^+$ by $\mathsf{B}$ and its elements are called words. We denote by $\mathsf{F}$ the set of all words such that each word is built up from both variables $0$ and $1$. Each element in $\mathsf{F}$ is called a full word.

Let $\mathbf{S}$ be an ordered semigroup, and $\alpha = \alpha_1 \cdots \alpha_m \in \mathsf{B}$ with $\vert \alpha \vert = m$ for some $m \in \mathbb{N}$. We define a valuation of $\alpha$ in $\mathcal{P}(S)$ as a mapping $\overline{\alpha} \colon \mathcal{P}(S) \to \mathcal{P}(S)$ assigning by $$\overline{\alpha}(A) := \begin{cases} \emptyset & \text{if } A = \emptyset, \\ B_1 \cdots B_m & \text{otherwise}, \end{cases} $$ where $$B_i = \begin{cases} A & \text{if } \alpha_i = 1, \\ S & \text{otherwise}, \end{cases} $$ for all $i \in [m]$.

Now, we are ready to recall the notion of $\alpha$-ideals in ordered semigorups. Let $\mathbf{S}$ be an ordered semigroup, and $\alpha \in \mathsf{F}$. A subsemigroup $A$ of $\mathbf{S}$ satisfying $(A] \subseteq A$ is said to be an $\alpha$-ideal of $\mathbf{S}$ if $$\overline{\alpha}(A) \subseteq A.$$ We see that an $(m,n)$-ideal and $n$-interior ideal is a $1^m01^n$-ideal and $01^n0$-ideal, respectively. In this case, we can regard the concept of $\alpha$-ideals a generalizations of $(m,n)$-ideals and $n$-interior ideals. We observe that the notion of $\alpha$-ideals sheds us to understand several unknown ideals in ordered semigroups.

On March, 2022, I was assigned by MathSciNet to review a paper written by Niovi Kehayopulu. The paper considered three related concepts: $poe$-semigroups, $\vee e$-semigroups, and $le$-semigroups.

A $poe$-semigroup is an ordered semigroup $\mathbf{S}$ possess the greatest element $e$. That is, $x \leq e$ for all $x \in S$. An algebra $\langle S; \cdot, \vee \rangle$ of type $(2,2)$ is said to be a $\vee e$-semigroup if $\langle S; \cdot \rangle$ is a semigroup, $\langle S; \vee \rangle$ is a $\vee$-semilattice with the greatest element $e$, and the operation $\cdot$ distributes over $\vee$. Finally, an algebra $\langle S; \cdot, \vee, \wedge \rangle$ is called an lattice ordered semigroup with the greatest element or $le$-semigroup if $\langle S; \cdot \rangle$ is a semigroup, $\langle S; \vee, \wedge \rangle$ is a lattice with the greatest element $e$, and the operation $\cdot$ distributes over $\vee$.

It is not difficult to see that we can regard a $\vee e$-semigroup and an $le$-semigroup as a $poe$-semigroup in a natural way. Indeed, the compatible property is induced by the distibutivity of $\cdot$ over $\vee$. This means that the above three structures are ordered semigroups.

Kehayopulu described the relationships of (hyper)ideals among these three related structures and others, for instance, (ordered) semigroup and (ordered) hypersemigroups. Moreover, the previous study of Kehayopulu illustrated that the concept of $poe$-semigroups is a representation of ordered semigroups. This study inspire me for further study ideals in the context of these three structures.

Let $\mathbf{S}$ be a $poe$-semigroup, and $\alpha = \alpha_1 \cdots \alpha_m \in \mathsf{B}$ with $\vert \alpha \vert = m$ for some $m \in \mathbb{N}$. Let us modify the notion of valuation on $\mathcal{P}(S)$ as follows. The valuation of $\alpha$ on $S$ is defined as a mapping $\overline{\alpha} \colon S \to S$ assigning by $\overline{\alpha}(a) := x_1 \cdots x_m$, where $$x_i = \begin{cases} a & \text{if } \alpha_i = 1, \\ e & \text{otherwise}, \end{cases} $$ for all $i \in [m]$. By this defining, we see that for every $\alpha, \beta \in \mathsf{B}$, $\overline{\alpha \beta}(a) = \overline{\alpha}(a) \overline{\beta}(a)$ for all $a \in S$.

Let $\mathbf{S}$ be a $poe$-semigroup. An element $a \in S$ is said to be a subidempotent of $\mathbf{S}$ if $a^2 \leq a$. Given $\alpha \in \mathsf{F}$, a subidempotent $a$ of $\mathbf{S}$ is called an $\alpha$-ideal element of $\mathbf{S}$ if $$\overline{\alpha}(a) \leq a.$$ Now, let us briefly illustrate the relation between the concepts of $\alpha$-ideals in ordered semigroups and $\alpha$-ideal elements in $poe$-semigroups.

Let $\mathbf{S}$ be an ordered semigroup. Then, $\mathbf{S}$ induces an algebraic system $\mathbf{P(S)} := \langle \mathcal{P}(S); \widehat{\cdot}, \subseteq \rangle$. Here we note that the operation $\widehat{\cdot}$ is the extension of $\cdot$ defined above. One could examine that $\mathbf{P(S)}$ is a $poe$-semigroup with $e := S$. By this configulations, we observe that any $\alpha$-ideal of $\mathbf{S}$ is an $\alpha$-ideal element of $\mathbf{P(S)}$.

Now, we know that the notion of $\alpha$-ideal elements of $poe$-semigroups abstracts $(m,n)$-ideals and $n$-interior ideals. We might ask that how about two-sided ideals, quasi-ideals, or even other type of ideals? Can the notion of full words extend to this? To extend this concept, we need some further investigations of full words.

Let $\alpha \in \mathsf{F}$, and $n \in \mathbb{N}$ such that $n \geq \vert \alpha \vert$. An $n$-tuple $(u_1, \ldots, u_n) \in \{\varepsilon,0,1\}^n$ is called a canonical tuple of $\alpha$ with length $n$ if $\alpha = u_1 \cdots u_n$ and the following statements hold: (1) $u_1 \not = \varepsilon$; (2) $u_i = \varepsilon$ implies $u_{i-1} \not = 1$ for all $1 < i \leq n$. We denote the set of all canonical tuple of $\alpha$ with length $n$ by $\mathsf{C}_n(\alpha)$. For instance, let $\alpha = 1010$. Then, $(1,0,1,0), (1,0,1,0,\varepsilon), (1,0,\varepsilon,1,0,\varepsilon)$ and $(1,0,1,0,\varepsilon,\varepsilon,\varepsilon)$ are canonical tuples of $\alpha$ with length $4,5,6$ and $7$, respective. However, $(1,0,1,\varepsilon,0)$ is not a canonical tuple of $\alpha$ with length $5$.

Let $C = \{\varepsilon,0,1\}$ and ${\leq_C} := {\Delta_C \cup \{ (\varepsilon,0), (\varepsilon,1), (0,1) \}}$, where $\Delta_C$ is the equality relation on $C$. Then, the structure $\mathbf{C} := \langle C; \leq_C \rangle$ is a totally ordered set. This structure plays an important role in comparing full words.

Let $\alpha,\beta \in \mathsf{F}$ and $n \in \mathbb{N}$ such that $n \geq \max\{ \vert \alpha \vert, \vert \beta \vert \}$. Suppose that $\mathbf{u} = (u_1,\ldots,u_n) \in \mathsf{C}_n(\alpha)$ and $\mathbf{v} = (v_1,\ldots,v_n) \in \mathsf{C}_n(\beta)$. We say that $\mathbf{u}$ is a reduction of $\mathbf{v}$ if $u_i \leq_C v_i$ for all $i \in [n]$. In this case, we write $\mathbf{u} \leq \mathbf{v}$, otherwise $\mathbf{u} \not \leq \mathbf{v}$. For example, let $\alpha = 1010$ and $\beta = 10110$. Suppose that $\mathbf{u} = (1,0,\varepsilon,1,0,\varepsilon), \mathbf{v} = (1,0,1,0,\varepsilon,\varepsilon)$ and $\mathbf{w} = (1,0,\varepsilon,1,1,0,\varepsilon)$. Then, $\mathbf{u}, \mathbf{v} \in \mathsf{C}_6(\alpha)$ and $\mathbf{w} \in \mathsf{C}_6(\beta)$. We see that $\mathbf{u} \leq \mathbf{w}$, but $\mathbf{u} \not \leq \mathbf{w}$.

Now, let us introduce the concept of the reduction relation on the set of all full words. Let $\alpha,\beta \in \mathsf{F}$. We say that $\alpha$ is a reduction of $\beta$, denoted by $\alpha \leq \beta$, if for any $n \in \mathbb{N}$ with $n \geq \vert \beta \vert$ and $\mathbf{v} \in \mathsf{C}_n(\beta)$, there exists $\mathbf{u} \in \mathsf{C}_n(\alpha)$ such that $\mathbf{u} \leq \mathbf{v}$. Otherwise, we say that $\alpha$ is not a reduction of $\beta$ and we denote by $\alpha \not \leq \beta$. For example, $10 \leq 101$, but $010 \not \leq 101$.

We observe that considering whether two full words are related under the reduction relation is not convenient. In the 18th International Conference of Young Algebraists (ICYAT2025) in Thailand, Erkko Lehtonen suggested me the following criteria for determining this relation as follows.

Proposition Let $\alpha, \beta \in \mathsf{F}$. Then, $\alpha \leq \beta$ if and only if there exists $\mathbf{u} \in \mathsf{C}_{\vert \beta \vert}(\alpha)$ such that $\mathbf{u} \leq \mathbf{v}$, where $\mathbf{v} \in \mathsf{C}_{\vert \beta \vert}(\beta)$.

Collecting all results of the reduction relation, we obtain the following important result.

Theorem For $\circ_{\mathrm{con}}$ the concatenation of words and $\leq$ the reduction relation, we have that the structure $\mathbf{F} := \langle \mathsf{F}; \circ_{\mathrm{con}}, \leq \rangle$ is an ordered semigroup.

The concept of the reduction relation not only helps us define an abstraction of $\alpha$-ideal elements in $poe$-semigroups, but also plays a crucial role in the investigation of several algebraic properties of $poe$-semigroups.

Let $X$ be a nonempty finite antichain of $\mathbf{F}$. A partition of $X$ is called an ideal assignment set of $X$, shortly an ideal assignment set. If it is a trivial partition, we call a trivial ideal assignment set. For instance, $\{ \{10\}, \{01,010\} \}, \{ \{10,0101\} \}$ are ideal assignment sets. Indeed, $\{ \{10,0101\} \}$ is a trivial ideal assignment set. However, $\{ \{10,101\}, \{010\} \}$ is not an ideal assignment set.

Now, we are ready to define an abstraction of $\alpha$-ideal elements. Let $\mathbf{S}$ be an $le$-semigroup, and $\mathcal{A}$ an ideal assignment set. A subidempotent $a$ of $\mathbf{S}$ is said to be an $\mathcal{A}$-ideal element of $\mathbf{S}$ if $$\bigwedge_{\alpha \in \Gamma} \overline{\alpha}(a) \leq a$$ for every block $\Gamma$ of $\mathcal{A}$.

We observe that:

The concept of partition ideal elements is a satisfactory concept that can describe the known notions of ideal elements. Furthermore, this concept sheds light on understanding several unknown ideal elements. This could be a beneficial tool in the investigation of ordered semigroup theory. Therefore, in my focus, I am considering several algebraic properties of ordered semigroups in terms of partition ideal elements.