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 R with the help of an ordered multiplicative group Γ with a zero adjoined. Valente [23] examined the interplay between the orderings of Γ and valuations of R. Valuation rings, defined from valuations, were first studied for fields. While studying them for noncommutative rings, Schilling [21] considered valuations of a division ring to a simply ordered l-group. The author studied the different ideals associated with this valuation, the valuation ring, and the valuation on a quotient ring associated with the valuation. Schilling also initiated the study of invariant valuation rings of division rings. Gubareni [11] continued with the study of valuations of division rings. Nasehpour [20] defined valuation maps from a semiring to a totally ordered commutative monoid.

Definition 1.1.([11]) A totally ordered (or linearly ordered) group is an additive group G along with a binary order relation ≥ which satisfies the following axioms:

• (i) if α≥β and β≥α then α=β;

• (ii) if α≥β and β≥γ then α≥γ;

• (iii) if α≥β then γ+α≥γ+β and α+γ≥β+γ; and

• (iv) either α≥β or β≥α,

for any α,β,γ∈G.

Let us consider a totally ordered (additive) group G. We now add an element ∞ which is larger than any other element of G which satisfies the condition

and denote G∪{∞} by G^*.

Definition 1.2.([24]) Let K' be the multiplicative group of a field K. Let Γ be an additive abelian group which is totally ordered. A valuation of K is a mapping of v:K′→Γ which satisfies the following conditions:

• (i) v(xy)=v(x)+v(y),

• (ii) v(x+y)≥min{v(x),v(y)}.

Given such a valuation v, the element v(x)∈Γ, for x in K', is called the value of x. The set of values of elements of K' forms a subgroup of Γ which is said to be the value group of v. It is assumed that the mapping v is onto Γ. Condition (i) signifies that v is a group homomorphism of K' (a multiplicative group) onto Γ (an additive group). Thus v(1)=0. Further 1=(-1)(-1) yields v(-1)=0. We now obtain v(-x)=v(x). The consequences that follow are [24]:

• (i) v(x−y)≥min{v(x),v(y)},

• (ii) v(y/x)=v(y)−v(x),x≠0,

• (iii) v(1/x)=−v(x),x≠0,

• (iv) v(x)<v(y)⇒v(x+y)=v(x).

The set {x∈K:v(x)≥0} forms a ring, denoted by Rv and is said to be the valuation ring of v. For every x from K, either x or 1/x is an element of the valuation ring. This happens because either v(x)≥0 or v(x)≤0 and so v(x)≥0 or v(1/x)≥0. Both v(x) and -v(x) being nonnegative implies they are 0, so the kernel of the homomorphism v gives us the multiplicative group of units in Rv. The set Pv={y∈K:v(y)>0} is the set of nonunits in Rv. We find that Pv is not only a prime ideal but also a maximal ideal of Rv. The divisibility relation y|x in Rv is the same as the relation v(x)≥v(y).

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 k-valuation semirings.

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 S equipped with a binary operation called addition and a ternary multiplication (denoted by juxtaposition) is said to be a ternary semiring if S is an additive commutative semigroup which satisfies for all a, b, c, d, e ∈ S

• (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 S, we mean an element 0 in S, provided such an element exists, which satisfies 0+x=x and 0xy=0=x0y=xy0 for all x,y in S. If one exists, then S is called a ternary semiring with zero.

Example 1.4. The set of all non-positive integers, ℤ0− with respect to usual (binary) addition and ternary multiplication is a ternary semiring.

Example 1.5. Let us consider a topological space X. Then the set S consisting of continuous functions f:X→ℝ−, where ℝ− stands for the set of negative real numbers, is a ternary semiring in which binary addition and ternary multiplication are defined by

for all f,g,h ∈ S and x ∈ X.

Two more examples of ternary semiring are S1={r2:r∈ℚ} and S2={ri:r∈ℝ} both with respect to usual (binary) addition and ternary multiplication.

A ternary semiring S is called additively cancellative if a+b=a+c implies that b=c, for any a,b,c in S. By a ternary subsemiring of S we mean an additive subsemigroup T of S in which abc ∈ T for any a,b,c ∈ T. We call an additive subsemigroup I of S a left ideal of S if SSI⊆I, a lateral ideal if SIS⊆I and a right ideal if ISS⊆I. If I is both a left ideal as well as a right ideal of S then it is called a (two sided) ideal of S. If I is further a lateral ideal then it is called an ideal of S.

An ideal I of S is a k-ideal if x+y ∈ I where x ∈ S and y ∈ I yield x ∈ I. A prime ideal is a proper ideal P of S such that if ABC ⊆ P then either A ⊆ P or B⊆P or C ⊆ P, where A,B,C are any three ideals of S. It can be proved that if S is commutative, an equivalent condition for a proper ideal P of S to be prime is that abc ∈ P implies that either a ∈ P or b ∈ P or c ∈ P. An element a ∈ S is said to be invertible in S if one can find an element b ∈ S (called the ternary semiring inverse of a) satisfying abx=xab=bax=xba=x,∀x∈S. If |S|≥2 then S is called a ternary division semiring if every non-zero element of S is invertible. If there is an element e ∈ S such that eex=exe=xee=x for all x ∈ S, then it is called a unital element of S. In this case exy=xey=xye for all x,y. For example, ℤ0− is a ternary semiring with 0 as zero element and -1 as unital element. Again, let k be a fixed number in ℝ. If we define a+b=0 and abc=a+b+c+k for all a,b,c∈ℝ, then -k/2 is a unital element in the ternary semiring ℝ. A ternary semiring may contain more than one unital element. Consider the ternary semiring ℚ of rational numbers with respect to ordinary addition and ternary multiplication [ ] defined by [abc]=abc, for all a,b,c∈ℚ. Here, 1 and -1 are both unital elements.

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.

We consider a commutative ternary semiring S with a unital element e and a zero element 0, |S|≥2. Let us introduce the notion of valuation on S as follows.

Definition 2.1. A valuation on S is a function v:S→G* fulfilling the following conditions:

• (i) v(abc)=v(a)+v(b)+v(c),

• (ii) v(a+b)≥min{v(a),v(b)},

• (iii) v(e)=0, and

• (iv) v(0)=∞,

for all a,b,c∈S.

Suppose S is a ternary semiring with only two elements, a unital element e and a zero element 0. Let G be any totally ordered group. Then v:S→G* defined by

is always a valuation on S. We call this function a trivial valuation.

Example 2.2. We consider the ternary semiring S=ℤ0− of Example 1.4 and let G be the group (ℝ,+) totally ordered by usual ≤, G*=G∪{∞}. Let v:S→G* be a function defined by

Example 2.3. Let S be the ternary semiring of all polynomial functions on X with coefficients from ℝ− (we refer to Example 1.5) where X=[0,∞) and G=(ℝ,+). We define a function v:S→G* by

if f(x)=a0+a1x+a2x2+⋯+amxm

Then v is a valuation on S.

We note that in a ternary semiring S containing the additive inverses of its elements, the following equality holds for any a,b,c ∈ S:

So we find that -e, if present in S, is a unital element of S if e is. In fact, for all x ∈ S, eex=x yields x=-(-x)=-(-(eex))=-((-e)ex)=(-e)(-e)x. We further obtain the following.

Lemma 2.4. Let S be a ternary semiring, G be a totally ordered group, and v:S→G* be a valuation on S. Then

• (i) if e is a unital element of S, v(-e)=0, provided -e ∈ S.

• (ii) v(-a)=v(a), for all a ∈ S with -a ∈ S.

Proof. (i) As -e is a unital element of S, v(-e)=0. (ii) Let a ∈ S and e be a unital element of S. Then a =(-e)(-e)a=(-e)e(-a) so that v(a) = v(-e)+v(e)+v(-a). Hence v(a)=v(-a).

Let val(S) denote the set of valuations of S. For v∈val(S) we define three subsets of S:

The sets Av and Iv are nonempty as v(e)=0 and v(0)=∞.

Theorem 2.5. Av is a ternary subsemiring of S.

Proof. It is clear that Av is an additively commutative subsemigroup of S. Let t1,t2,t3∈Av, then v(t1)≥0,v(t2)≥0,v(t3)≥0. Now

This implies that t1t2t3∈Av. Therefore Av becomes a ternary subsemiring of S.

Theorem 2.6. Suppose Pv is nonempty. Then Pv is an ideal of Av. Further, Pv is a k-ideal of Av if v(x)<v(y) implies v(x+y)=v(x).

Proof. Obviously Pv is an additive subsemigroup of S. Let x∈AvAvPv. Then x=a1a2a3 for some a1,a2∈Av and a3∈Pv. So

and x∈Pv. Therefore AvAvPv⊆Pv. Similarly we can show that AvPvAv⊆Pv and PvAvAv⊆Pv. Hence Pv is an ideal of Av.

Again, let x∈Av and y,x+y∈Pv. Then v(x)≥0,v(y)>0, and v(x+y)>0, so we have v(x+y)≥min{v(x),v(y)}. If v(x) equals v(y) then obviously x∈Pv.

Let us suppose that v(x)≠v(y). If min{v(x),v(y)}=v(x), then v(x)>0 otherwise if v(x) = 0, then v(x)<v(y) implies that v(x+y)=v(x)=0, a contradiction. So x∈Pv. If min{v(x),v(y)}=v(y), then v(x)≥v(y)>0 and we obtain x∈Pv. Therefore Pv is a k-ideal of Av.

Theorem 2.7. Iv is a prime ideal of S.

Proof. Undoubtedly Iv is an additive commutative subsemigroup of S. Let x=a1a2a3∈IvSS for some a1∈Iv and a1,a2∈S then v(a1)=∞ and

So x∈Iv. Thus IvSS⊆Iv. We can show in a similar way that SIvS⊆Iv and SSIv⊆Iv. Hence Iv is an ideal of S.

Let a, b, c be elements of S such that abc∈Iv. Then

This gives us at least one of v(a),v(b),v(c) to be ∞, i.e., at least one of a, b, c is in Iv. Therefore Iv is a prime ideal of S.

We now consider a ternary division semiring D containing a unital element e, with a valuation v as defined in Definition 2.1.

Example 3.1. Let

be the set of real quaternions. Then D is a ternary division semiring under binary addition and ternary multiplication defined as follows:

and

where dij=∑aipbpqcqj.

We define a mapping v:D→ℝ by

Then v becomes a valuation on D.

Let us denote by U(A), the set {u∈A:v(u)=0} for any ternary subsemiring A of D. For simplicity, we write U for U(D). Then 0 ∉ U and U is nonempty as v(e)=0 for a unital element e. Further if u1,u2,u3∈U then

and so u1u2u3∈U. Thus U is a ternary subsemiring of D.

Now we let D*=D∖{0}. Then D^* is a ternary multiplicative semigroup [22], i.e., the ternary multiplication satisfies associative property.

Theorem 3.2. Let D be a ternary division semiring. Then U is a ternary subsemigroup of D^* containing the unital element(s). Further, any ternary inverse of an element of U also belongs to U.

Proof. The first part is obvious. Let a ∈ U then v(a) = 0 and there exists an element b ∈ D such that abx = x = xab = bax = xba for all x ∈ D. Now

implies v(a)+v(b) = 0, or,v(a)=-v(b), or, v(b) = 0. So b ∈ U and our theorem is proved.

We call the ternary subsemigroup U the ternary semigroup of valuation units of D. We observe that there may be a unit a in D such that v(a) ≠ 0.

We now consider the subsets Av,Pv and Iv for D. Then we note that U(Av) is the ternary semigroup of valuation units of D. We obtain the following results in D.

Theorem 3.3. Pv is a maximal ideal of Av.

Proof. Let us suppose that I is an ideal of Av properly containing Pv. Let a∈Av. Now Pv=Av∖U. So there exists an element b∈U∩I. Then b∈U(Av) and so there is an element c∈Av such that bcx=xbc=cbx=xcb=x for all x∈Av. In particular, bca=a. Since IAvAv⊆I, a ∈ I. Thus Av=I and our contention is justified.

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

Definition 3.4. A ternary subsemiring A of a ternary division semiring D is called a ternary valuation semiring of D if for any element a(≠0)∈A, either a ∈ A or b ∈ A for any ternary inverse b of a in D.

Theorem 3.5. Let v be a valuation of a ternary division semiring D into a totally ordered abelian group G. Further let a ∈ D^* and b be a ternary inverse of a in D^*. Then

• (i) aAvb⊆Av, and

• (ii) either a∈Av or b∈Av, and thus Av is a ternary valuation semiring of D.

Proof. (i) For any x ∈ D, abx=x yields v(a)+v(b)=0. Now let x=ayb∈aAvb where y∈Av. Then we find that

and so x∈Av.

(ii) Suppose a ∈ D^* such that a∉Av, then v(a)<0. Also v(a)+v(b)=0 gives us v(b)=−v(a)>0. Thus b∈Av.

Theorem 3.6. In a ternary division semiring D, let a,b∈Av and we consider the following conditions:

• (i) a = bc₁c₂ where c1,c2∈Av,

• (ii) a = c₃c₄b where c3,c4∈Av.

If condition (i) or condition (ii) holds, then v(a)≥v(b). Further, if we assume that v(a)≥v(b) implies both conditions (i) and (ii), then any left ideal of Av is also a right ideal of Av and vice versa.

Proof. The proof of the first part is immediate. Now let I be a left ideal of Av, i.e., AvAvI⊆I. Let x=x1x2x3∈IAvAv for some x₁ ∈ I and x2,x3∈Av. Then

By our assumption x=c3c4x1 where c3,c4∈Av. Then x∈AvAvI⊆I. Thus IAvAv⊂I and so I is a right ideal of Av.

We will now consider a ternary semiring S valued epimorphically upon a totally ordered group G by the valuation function v. We find that a subgroup H of a totally ordered group G is called convex if α∈H,β≤α yields β ∈ H. We define convexity in a ternary semiring by employing v.

Definition 4.1. A subset T of a ternary semiring S is called convex if

• (i) a∈T,v(b)≤v(a) implies b ∈ T, and

• (ii) a,b∈T,v(c)=v(a)+v(b) implies c ∈ T.

Theorem 4.2. Let the ternary semiring S be valued epimorphically by v onto the totally ordered group G. If T is a convex subsemiring containing 0 of S then v(T) is a convex subsemigroup containing ∞ of G⋆.

Proof. Let α∈v(T) and β≤α in G. Then there exists a∈T,b∈S such that v(a)=α and v(b)=β. As a consequence of T being a convex subset of S, v(b)≤v(a) gives us b ∈ T and further v(b)=β∈v(T). Next let α=v(a),β=v(b) be elements in v(T) where a, b ∈ T. Let γ=α+β. As v is an epimorphism we obtain c ∈ S such that v(c)=γ=v(a)+v(b). Property (ii) of convexity of T yields c ∈ T. So v(c)=γ is in v(T). Since 0∈T,v(0)=∞∈v(T). Also if α=v(a),β=v(b) and γ=v(c) in v(T) where a,b,c are in T, then α+(β+γ)=v(a)+(v(b)+v(c))=v(abc)=(v(a)+v(b))+v(c)=(α+β)+γ. Hence v(T) is an additive convex semigroup of G⋆.

Theorem 4.3. Let v be defined as in Theorem 4.2. If H is a convex subsemigroup containing ∞ of G^* thenT=v−1(H) is a ternary convex subsemiring of S.

Proof. T≠ϕ is obvious. Let a, b, c∈ T then v(a),v(b),v(c)∈H. We find that v(a+b) ∈ H for v(a+b)=min{v(a),v(b)}, and so a+b ∈ T. Again v(abc)=v(a)+v(b)+v(c)∈H and we obtain abc ∈ T. Thus T is a ternary subsemiring of S. Further let a∈T,b∈S and v(b)≤v(a). Then v(a) ∈ H. As H is convex, we obtain v(b) ∈ H which yields b ∈ T. Lastly for a, b ∈ T, we note that v(a)+v(b)∈H. Let v(a)+v(b)=v(c) for some c ∈ S. So v(c)∈ H and c ∈ T. Therefore T is a convex ternary subsemiring of S.