In the paper, commutative semigroups simple over their endomorphism semirings are investigated. In particular, commutative semigroups having just two fully invariant congruences are classified into five basic types and each of these types

In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zeromultiplication rings of prime order, matrix rings over finite fields, ones with trivial addition and those that are additively idempotent.

We consider weighted tree automata with discounting over commutative semirings. For their behaviors we establish a Kleene theorem and an MSOlogic characterization. We introduce also weighted Muller tree automata with discounting over the max-plus and the min-plus semirings, and we show their expressive equivalence with two fragments of weighted MSO-sentences.

The Cayley-Hamilton theorem (CHT) is a classic result in linear algebra over fields which states that a matrix satisfies its own characteristic polynomial. CHT has been extended from fields to commutative semirings by Rutherford in 1964. However, to the best of our knowledge, no result is known for noncommutative semirings. This is a serious limitation, as the class of regular languages, with f...

In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zeromultiplication rings of prime order, matrix rings over finite fields, and those that are additively idempotent.

Since the theory of ideals plays an important role in the theory of semirings, in this paper we will make an intensive study of the notions of primal and weakly primal ideals in commutative semirings with an identity 1. It is shown that these notions inherit most of the essential properties of the primal and weakly primal ideals of a commutative ring with non-zero identity. Also, the relationsh...

Though more general definitions are sometimes used, for this paper a semiring will be defined to be a set S with two operations + (addition) and 9 (multiplication); with respect to addition, S is a commutative monoid with 0 as its identity dement 9 With respect to multiplication, S is a (generaily noncommutat ive) monoid with 1 as its identity element. Connecting the two algebraic structures ar...

The relationship between MV-algebras and semirings was described by A. Di Nola and B. Gerla ([7]). Since commutative basic algebras are similar to MV-algebras up to associativity of the binary operation we try to get a similar relationship between commutative basic algebras and so-called near semirings and we show that this is possible. This means that associativity does not play an important r...

Abstract In this paper we develop a number of results and notions concerning Positivstellensätze for semirings (preprimes) commutative unital real algebras. First reduce the Archimedean Positivstellensatz to corresponding result quadratic modules. Various applications are investigated. A general with denominators is proved filtered algebras semirings. As an application derive denominator-free c...