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REGULAR Γ−INCLINE and FIELD Γ−SEMIRING 1. Introduction

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REGULAR Γ−INCLINE and FIELD Γ−SEMIRING 1. Introduction Novi Sad J. Math. Vol. 45, No. 2, 2015, 155-171 REGULAR Γ−INCLINE AND FIELD Γ−SEMIRING M. Murali Krishna Rao1 and B. Venkateswarlu2 Abstract. We introduce the notion of Γ−incline as a generalization of incline, which is an algebraic structure with an additional poset struc- ture. We introduce the notions of regular Γ−incline, integral Γ−incline, field Γ−incline, field Γ−semiring, simple Γ−semiring and pre-integral Γ−semiring. We study their properties and relations between them. We prove that if M is a linearly ordered regular Γ−incline, then M is a com- mutative Γ−incline and M is a field Γ−semiring with an additional prop- erty if and only if M is an integral, simple and commutative Γ−semiring. AMS Mathematics Subject Classification (2010): 06F25, 16A09 Key words and phrases: Γ−semiring; Γ−incline; regular Γ−incline; inte- gral Γ−incline; field Γ−incline; field Γ−semiring; simple Γ−semiring 1. Introduction The notion of a semiring was introduced by H. S. Vandiver [15] in 1934. Semiring is a well known universal algebra. Semirings have been used for studying optimization theory, graph theory, matrices, determinants, theory of automata, coding theory, analysis of computer programs, etc. The notion of a Γ-ring was introduced by N. Nobusawa [13] as a generaliza- tion of the rings in 1964. M. K. Sen [14] introduced the notion of Γ- semigroup in 1981. The notion of ternary algebraic system was introduced by Lehmer [7] in 1932, Lister [8] introduced ternary ring. Dutta & Kar [4] introduced the notion of ternary semiring which is a generalization of ternary ring and semir- ing. In 1995, M. Murali Krishna Rao [11] introduced the notion of Γ- semiring which is a generalization of Γ- ring, ternary semiring and semiring. After the paper [11] was published, many mathematicians obtained interesting results on Γ-semirings. The concept of an incline was first introduced by Z. Q. Cao [3] in 1984. Inclines are additively idempotent semirings in which products are less than or equal to either factor. Products reduce the values of quantities and make them go down therefore the structures were named inclines. Idempotent semirings and Kleene algebras have recently been established as fundamental structures in computer sciences. An incline is a generalization of Boolean algebra, fuzzy algebra and distributive lattice and incline is a special type of semiring. An incline has both the semiring structure and the poset structure. Every dis- tributive lattice and every Boolean algebra is an incline but an incline need not 1Department of Mathematics, GIT, GITAM University, Visakhapatnam-530 045, Andhra Pradesh, India. e-mail: [email protected] 2Department of Mathematics, GIT, GITAM University, Visakhapatnam-530 045, Andhra Pradesh, India. e-mail: [email protected] 156 M. Murali Krishna Rao and B. Venkateswarlu be a distributive lattice. The set of all idempotent elements in an incline is a distributive lattice. Von Neumann [12] introduced the concept of regular ele- ments in a ring. A. R. Meenakshi and S. Anbalagan [9] studied regular elements in an incline and proved that a regular commutative incline is a distributive lattice. An incline is a more general algebraic structure than a distributive lat- tice. Ahn, Jun and Kim [1] have proved that \Every prime ideal of the incline M is equivalent to an irreducible ideal of M". In an incline every ideal is a k-ideal. A. R. Meenakshi and N. Jayabalan [10] have proved that every prime ideal in the incline M is an irreducible ideal of M and every maximal ideal in the incline M is an irreducible ideal. W. Yao and S. C. Han [16] studied the relations between ideals, filters and congruences in inclines and it is shown that there is a one to one correspondence between the set of ideals and the set of all regular congruences. Z. Q. Cao et al. [2] studied the incline and its appli- cations. Kim and Rowsh [5, 6] have studied matrices over an incline. Many research scholars have researched the theory of incline matrices. Few research scholars studied the algebraic structure of inclines. Inclines and matrices over inclines are useful tools in diverse areas such as automata theory, design of switching circuits, graph theory, information systems, modeling, decision mak- ing, dynamical programming, control theory, classical and nonclassical path finding problems in graphs, fuzzy set theory, data analysis, medical diagnosis, nervous system, probable reasoning, physical measurement and so on. In this paper, we introduce the notion of Γ−incline as a generalization of the incline, which is simultaneously an algebraic structure and a poset structure. We introduce the notions of regular Γ−incline, idempotent Γ−incline, integral Γ−incline, pre-integral Γ−incline, mono Γ−incline, field Γ−incline and study the relations between them and their properties. Also, we introduce the notion of field Γ−semiring, simple Γ−semiring, pre-integral Γ−semiring and integral Γ−semiring and we study the relations between them and their properties. 2. Preliminaries In this section we will recall some of the fundamental concepts and defini- tions, which are necessary for this paper. Definition 2.1. [2] A set S together with two associative binary operations called addition and multiplication (denoted by + and ·, respectively) will be called a semiring, provided (i) addition is a commutative operation. (ii) multiplication distributes over addition both from the left and from the right. (iii) there exists 0 2 S such that x + 0 = x and x · 0 = 0 · x = 0 for each x 2 S: Definition 2.2. [3] A commutative incline M with additive identity 0 and multiplicative identity 1 is a non-empty set M with operations addition (+) and multiplication (:) defined on M × M ! M satisfying the following conditions for all x; y; z 2 M Regular Γ−incline and field Γ−semiring 157 (i) x + y = y + x (ii) x + x = x (iii) x + xy = x (iv) x + (y + z) = (x + y) + z (v) x(yz) = x(yz) (vi) x(y + z) = xy + xz (vii) (x + y)z = xz + yz (viii) x1 = 1x = x (ix) x + 0 = 0 + x = x (x) xy = yx Definition 2.3. [13] Let M and Γ be additive abelian groups. If there exists a mapping M × Γ × M ! M (images of (x; α; y) will be denoted by xαy; x; y 2 M; α 2 Γ) satisfying the following conditions for all x; y; z 2 M; α; β 2 Γ (i) xα(yβz) = (xαy)βz (ii) xα(y + z) = xαy + xαz (iii) x(α + β)y = xαy + xβy (iv) (x + y)αz = xαz + yαz: Then M is called a Γ− ring. Definition 2.4. [11] Let (M; +) and (Γ; +) be commutative semigroups. Then we call M a Γ−semiring, if there exists a mapping M × Γ × M ! M (images of (x; α; y) will be denoted by xαy; x; y 2 M; α 2 Γ) such that it satisfies the following axioms for all x; y; z 2 M and α; β 2 Γ (i) xα(y + z) = xαy + xαz (ii) (x + y)αz = xαz + yαz (iii) x(α + β)y = xαy + xβy (iv) xα(yβz) = (xαy)βz: Every semiring R is a Γ−semiring with Γ = R and ternary operation xγy defined as the usual semiring multiplication. We illustrate the definition of Γ−semiring by the following example. 158 M. Murali Krishna Rao and B. Venkateswarlu Example 2.5. Let S be a semiring and Mp;q(S) denote the additive abelian semigroup of all p × q matrices whose entries are from S: Then Mp;q(S) is a Γ−semiring with Γ = Mp;q(S) and the ternary operation defined by the usual matrix multiplication as xαy = x(αt)y; where αt denotes the transpose of the matrix α; for all x; y and α 2 Mp;q(S): Definition 2.6. [11]AΓ−semiring M is said to have a zero element if there exist an element 0 2 M such that 0 + x = x = x + 0 and 0αx = xα0 = 0; for all x 2 M; α 2 Γ: Definition 2.7. [11]AΓ−semiring M is said to be a commutative Γ−semiring if xαy = yαx; for all x; y 2 M and α 2 Γ: Definition 2.8. [11] Let M be a Γ−semiring. An element a 2 M is said to be α idempotent if a = aαa; an element of M is said to be an idempotent of M if it is α idempotent for some α 2 Γ: Definition 2.9. [11] Let M be a Γ−semiring. If every element of M is an idempotent of M; M is said to be an idempotent Γ−semiring. Definition 2.10. [11] Let M be a Γ−semiring. An element a 2 M is said to be a regular element of M if there exist x 2 M; α; β 2 Γ such that a = aαxβa: Definition 2.11. [11] Let M be a Γ−semiring. If every element of M is regular, then M is said to be a regular Γ−semiring. Definition 2.12. [11] A non-empty subset A of the Γ−semiring M is called a Γ−subsemiring M if (A; +) is a subsemigroup of (M; +) and aαb 2 A for all a; b 2 A and α 2 Γ: Definition 2.13. [11] An additive subsemigroup I of a Γ−semiring M is said to be a left (right) ideal of M if MΓI ⊆ I (IΓM ⊆ I): If I is both a left and right ideal then I is called an ideal of Γ−semiring M: 3. Regular Γ−incline In this section, we introduce the notions of Γ−incline, zero divisor, unity element, invertible element, regular element and idempotent element in Γ− in- cline and also the concepts of regular Γ−incline, integral Γ−incline, idempotent Γ−incline and field Γ−incline.
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