Commutative Semiring Research Articles

Some properties of fuzzy sets of type 2

The concept of fuzzy sets of type 2 has been defined by L. A. Zadeh as an extension of ordinary fuzzy sets. The fuzzy set of type 2 can be characterized by a fuzzy membership function the grade (or fuzzy grade) of which is a fuzzy set in the unit interval [0, 1] rather than a point in [0, 1]. This paper investigates the algebraic structures of fuzzy grades under the operations of join ⊔, meet ⊔, and negation ┐ which are defined by using the extension principle, and shows that convex fuzzy grades form a commutative semiring and normal convex fuzzy grades form a distributive lattice under ⊔ and ⊓. Moreover, the algebraic properties of fuzzy grades under the operations and which are slightly different from ⊔ and ⊓, respectively, are briefly discussed.

Monic and monic free ideals in a polynomial semiring in several variables

The study of monic and monic free ideals in a polynomial semiring S [ x ] S[x] , where S S is a commutative semiring with an identity, is extended to S [ x 1 , x 2 , … , x n ] S[{x_1},{x_2}, \ldots ,{x_n}] . Structure theorems are given for monic, monic free, and monic free k k -ideals in S [ x 1 , x 2 , … , x n ] S[{x_1},{x_2}, \ldots ,{x_n}] . It is shown that each monic free k k -ideal in S [ x 1 , … , x n ] , S S[{x_1}, \ldots ,{x_n}],\;S a strict semiring, is the sum of a finite number of ideals B i {B_i} such that each B i {B_i} is the union of a proper infinite ascending chain of ideals.

Monic and monic free ideals in a polynomial semiring

Two classes of ideals are introduced in a polynomial semiring S [ x ] S[x] , where S S is a commutative semiring with an identity. A structure theorem is given for each class.

The semiring of quotients of commutative semirings

The semiring of quotients of commutative semirings

XIX.—The Cayley-Hamilton Theorem for Semi-rings

SynopsisA new proof of the Cayley-Hamilton theorem avoiding the use of determinants makes it possible to apply this theorem to matrices over a commutative semi-ring. The relationship of this theorem to a theorem by Lunts concerning switching matrices is investigated.