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We prove here an Eilenberg type theorem: the so-called conjunctive varieties of rational languages correspond to the pseudovarieties of finite semilattice-ordered monoids. Taking complements of members of a conjunctive variety of languages we get a so-called disjunctive variety. We present here a non-trivial example of such a variety together with an equational characterization of the corresponding pseudovariety.

Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let ${\Gamma}_{I}\left(R\right)$ be a graph with the set of vertices ${S}_{I}\left(R\right)=\{x\in R\setminus I:x+y\in I$ for some $y\in R\setminus I\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in I$. We look at the diameter and girth of this graph. Also we discuss when ${\Gamma}_{I}\left(R\right)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.

We show that the validity of Parikh’s theorem for context-free languages depends only on a few equational properties of least pre-fixed points. Moreover, we exhibit an infinite basis of $\mu $-term equations of continuous commutative idempotent semirings.

We show that the validity of Parikh's theorem for context-free languages depends only on a few equational properties of least pre-fixed points. Moreover, we exhibit an infinite basis of μ-term equations of continuous commutative idempotent semirings.

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G\left(R\right)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...

Many infinite finitely generated ideal-simple commutative semirings are additively idempotent. It is not clear whether this is true in general. However, to solve the problem, one can restrict oneself only to parasemifields.

We show in an additive inverse regular semiring $(S,+,\xb7)$ with ${E}^{\u2022}\left(S\right)$ as the set of all multiplicative idempotents and ${E}^{+}\left(S\right)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e,f\in {E}^{\u2022}\left(S\right)$, $ef\in {E}^{+}\left(S\right)$ implies $fe\in {E}^{+}\left(S\right)$. (ii) $(S,\xb7)$ is orthodox. (iii) $(S,\xb7)$ is a semilattice of groups. This result generalizes the corresponding result of regular ring.

Nous continuons dans ce second article, l’étude des outils algébrique de l’algèbre de la caractéristique 1 : nous examinons plus spécialement ici les algèbres de polynômes sur un semi-corps idempotent. Ce travail est motivé par le développement de la géométrie tropicale qui apparaît comme étant la géométrie algébrique de l’algèbre tropicale. En fait l’objet algébrique le plus intéressant est l’image de l’algèbre de polynôme dans son semi-corps des fractions. Nous pouvons ainsi retrouver sur les...

In this paper, specifically, we look at the preservation of the diameter and girth of the zero-divisor graph with respect to an ideal of a commutative ring when extending to a finite direct product of commutative rings.

Here we introduce the k-bi-ideals in semirings and the intra k-regular semirings. An intra k-regular semiring S is a semiring whose additive reduct is a semilattice and for each a ∈ S there exists x ∈ S such that a + xa²x = xa²x. Also it is a semiring in which every k-ideal is semiprime. Our aim in this article is to characterize both the k-regular semirings and intra k-regular semirings using of k-bi-ideals.

Commutative congruence-simple semirings were studied in [2] and [7] (but see also [1], [3]--[6]). The non-commutative case almost (see [8]) escaped notice so far. Whatever, every congruence-simple semiring is bi-ideal-simple and the aim of this very short note is to collect several pieces of information on these semirings.