### Article

Kyungpook Mathematical Journal 2022; 62(1): 57-67

**Published online** March 31, 2022

Copyright © Kyungpook Mathematical Journal.

### Valuations on Ternary Semirings

Sumana Pal^{*}, Jayasri Sircar, Pinki Mondal

Department of Mathematics and Statistics, Aliah University, Kolkata 700160, West Bengal, India

e-mail : sumana.pal@gmail.com

Department of Mathematics, Lady Brabourne College, Kolkata 700017, West Bengal, India

e-mail : jayasrisircar@gmail.com

Sunity Academy, Cooch Behar, West Bengal 736170, India

e-mail : pinkimondal1992@gmail.com

**Received**: October 6, 2020; **Revised**: September 1, 2021; **Accepted**: September 6, 2021

### Abstract

In the present study, we introduce a valuation of ternary semiring on an ordered abelian group. Motivated by the construction of valuation rings, we study some properties of ideals in ternary semiring arising in connection with the valuation map. We also explore ternary valuation semirings for a noncommuative ternary division semiring. We further consider the notion of convexity in a ternary semiring and how it is reflected in the valuation map.

**Keywords**: Valuation, valuation ring, ternary semiring, ternary division semiring

### 1. Introduction

Introduced by W. Krull [17] on fields in 1932, valuations on various algebraic systems have since been studied by many mathematicians. A valuation is a mapping into a field of real numbers, a linearly ordered group, or a linearly ordered semigroup. There are many notions of valuations on a commutative field or ring. Though valuation theory was initially connected precisely with commutative fields, later it was studied in noncommutative cases also.

Krull introduced valuations for rings in [18], and this has proved to be a very useful tool in ring theory, presenting a nice connection between rings and ordered abelian groups. The notion of valuation on a ring was covered by Bourbaki [1] in 1964. Manis [19] defined a valuation on a commutative ring

**Definition 1.1.**([11]) A totally ordered (or linearly ordered) group is an additive group

(i) if

$\alpha \ge \beta $ and$\beta \ge \alpha $ then$\alpha =\beta $ ;(ii) if

$\alpha \ge \beta $ and$\beta \ge \gamma $ then$\alpha \ge \gamma $ ;(iii) if

$\alpha \ge \beta $ then$\gamma +\alpha \ge \gamma +\beta $ and$\alpha +\gamma \ge \beta +\gamma $ ; and(iv) either

$\alpha \ge \beta $ or$\beta \ge \alpha $ ,

for any

Let us consider a totally ordered (additive) group

and denote ^{*}

**Definition 1.2.**([24]) Let

(i)

$v(xy)=v(x)+v(y)$ ,(ii)

$v(x+y)\ge \mathrm{min}\{v(x),v(y)\}$ .

Given such a valuation

(i)

$v(x-y)\ge \mathrm{min}\{v(x),v(y)\}$ ,(ii)

$v(y/x)=v(y)-v(x),x\ne 0$ ,(iii)

$v(1/x)=-v(x),x\ne 0$ ,(iv)

$v(x)<v(y)\Rightarrow v(x+y)=v(x)$ .

The set

The notion of a semiring was introduced by Vandiver in 1934 and since then the theory concerning semirings has evolved in various directions. It is well known that semirings have considerable applications not only in mathematics but also in computer science and operation research ([10], [12]). Nasehpour [20] explored valuation semirings in the realm of commutative semirings. Chang and Kim [2] worked on

Ternary semirings were introduced by Dutta and Kar in [6] and they continued this study in [7], [9], [14], and [15]. Many other authors, as well, have worked on ternary semirings, as one may find in references like [4], [8], [5], [3] and [13]. The concept of a ternary semiring has been applied to soft sets to introduce the notion of soft ternary semiring [16].

**Definition 1.3.**([6]) A nonempty set

(i)

$(abc)de=a(bcd)e=ab(cde)$ (associativity),(ii)

$ab(c+d)=abc+abd$ (left distributive law),(iii)

$(a+b)cd=acd+bcd$ (right distributive law), and(iv)

$a(b+c)d=abd+acd$ (lateral distributive law).

By a zero of the ternary semiring

**Example 1.4.** The set of all non-positive integers,

**Example 1.5.** Let us consider a topological space

for all

Two more examples of ternary semiring are

A ternary semiring

An ideal

The aim of this paper is to introduce a valuation on a ternary semiring and study the corresponding ideals which arise due to this valuation. We further discuss the basic properties of valuation ring in connection with a ternary division ring.

### 2. Valuation on Ternary Semiring

We consider a commutative ternary semiring

**Definition 2.1.** A valuation on

(i)

$v(abc)=v(a)+v(b)+v(c)$ ,(ii)

$v(a+b)\ge \mathrm{min}\{v(a),v(b)\}$ ,(iii)

v(e)=0 , and(iv)

$v(0)=\infty $ ,

for all

Suppose

is always a valuation on

**Example 2.2.** We consider the ternary semiring

**Example 2.3.** Let

if

Then

We note that in a ternary semiring

So we find that

**Lemma 2.4.** Let

(i) if

e is a unital element ofS ,v(-e)=0 , provided-e ∈ S .(ii)

v(-a)=v(a) , for alla ∈ S with-a ∈ S .

Let

The sets

**Theorem 2.5.**

This implies that

**Theorem 2.6.** Suppose

and

Again, let

Let us suppose that

**Theorem 2.7.**

So

Let

This gives us at least one of

### 3. Valuation on Noncommutative Ternary Semiring

We now consider a ternary division semiring

**Example 3.1.** Let

be the set of real quaternions. Then

and

where

We define a mapping

Then

Let us denote by

and so

Now we let ^{*}

**Theorem 3.2.** Let ^{*}

implies

We call the ternary subsemigroup

We now consider the subsets

**Theorem 3.3.**

The notion of valuation ring is now generalized for a ternary division ring.

**Definition 3.4.** A ternary subsemiring

**Theorem 3.5.** Let ^{*}^{*}

(i)

$a{\mathcal{A}}_{v}b\subseteq {\mathcal{A}}_{v}$ , and(ii) either

$a\in {\mathcal{A}}_{v}$ or$b\in {\mathcal{A}}_{v}$ , and thus${\mathcal{A}}_{v}$ is a ternary valuation semiring ofD .

and so

(ii) Suppose ^{*}

**Theorem 3.6.** In a ternary division semiring

(i)

a = bc where_{1}c_{2}${c}_{1},{c}_{2}\in {\mathcal{A}}_{v}$ ,(ii)

a = c where_{3}c_{4}b${c}_{3},{c}_{4}\in {\mathcal{A}}_{v}$ .

If condition (i) or condition (ii) holds, then

_{1} ∈ I

By our assumption

### 4. Convexity in Ternary Semiring

We will now consider a ternary semiring

**Definition 4.1.** A subset

(i)

$a\in T,v(b)\le v(a)$ impliesb ∈ T , and(ii)

$a,b\in T,v(c)=v(a)+v(b)$ impliesc ∈ T .

**Theorem 4.2.** Let the ternary semiring

**Theorem 4.3.** Let ^{*}

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