On algebraic congruence varieties over semirings

Abstract

In this paper, we develop some foundations for a theory of algebraic varieties of congruences on commutative semirings. By studying the structure of congruences, firstly, we show that the spectrum ${\text{Spec}}^c(A)$ consisting of prime congruences on a semirings $A$ has a Zariski topological structure; Then, for two semirings $A\subset B,$ we consider the polynomial semiring $S=A[x_1,\cdots ,x_n]$ and the affine $n-$space $B^n.$ For any congruence $\ddot{I}ƒ$ on $S$ and congruence $\ddot{I}$ on $B,$ we introduce the $\ddot{I}-$algebraic varieties $Z_{\ddot{I}}(\ddot{I}ƒ)(B)$ in $B^n,$ which are the set of zeros in $B^n$ of the system of polynomial $\ddot{I}-$congruence equations given by $\ddot{I}ƒ.$ When $\ddot{I}$ is a prime congruence, we find these varieties satisfying the axiom of closed sets, and forming a (Zariski) topology on $B^n.$ Some results about their structures including a version of Nullstellensatz of congruences are obtained.

34 pages