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 : firstname.lastname@example.org
Department of Mathematics, Lady Brabourne College, Kolkata 700017, West Bengal, India
e-mail : email@example.com
Sunity Academy, Cooch Behar, West Bengal 736170, India
e-mail : firstname.lastname@example.org
Received: October 6, 2020; Revised: September 1, 2021; Accepted: September 6, 2021
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
Introduced by W. Krull  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 , 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  in 1964. Manis  defined a valuation on a commutative ring
Definition 1.1.() A totally ordered (or linearly ordered) group is an additive group
and then ;
and then ;
then and ; and
Let us consider a totally ordered (additive) group
Definition 1.2.() Let
Given such a valuation
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 (, ). Nasehpour  explored valuation semirings in the realm of commutative semirings. Chang and Kim  worked on
Ternary semirings were introduced by Dutta and Kar in  and they continued this study in , , , and . Many other authors, as well, have worked on ternary semirings, as one may find in references like , , ,  and . The concept of a ternary semiring has been applied to soft sets to introduce the notion of soft ternary semiring .
Definition 1.3.() A nonempty set
(left distributive law),
(right distributive law), and
(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
Two more examples of ternary semiring are
A ternary semiring
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
is always a valuation on
Example 2.2. We consider the ternary semiring
Example 2.3. Let
We note that in a ternary semiring
So we find that
Lemma 2.4. Let
eis a unital element of S, v(-e)=0, provided -e ∈ S.
v(-a)=v(a), for all a ∈ Swith -a ∈ S.
This implies that
Theorem 2.6. Suppose
Let us suppose that
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
We define a mapping
Let us denote by
Now we let
Theorem 3.2. Let
We call the ternary subsemigroup
We now consider the subsets
The notion of valuation ring is now generalized for a ternary division ring.
Definition 3.4. A ternary subsemiring
Theorem 3.5. Let
or , and thus is a ternary valuation semiring of D.
Theorem 3.6. In a ternary division semiring
a = bc1c2where ,
a = c3c4bwhere .
If condition (i) or condition (ii) holds, then
By our assumption
4. Convexity in Ternary Semiring
We will now consider a ternary semiring
Definition 4.1. A subset
implies b ∈ T, and
implies c ∈ T.
Theorem 4.2. Let the ternary semiring
Theorem 4.3. Let
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