On algebraic congruence varieties over semirings

TL;DR

This paper develops a foundational theory of algebraic varieties of congruences over semirings, establishing topological and geometric structures analogous to classical algebraic geometry.

Contribution

It introduces the concept of algebraic varieties of congruences over semirings, including the Zariski topology and Nullstellensatz analogues for these structures.

Findings

Spectrum of prime congruences has a Zariski topology.

Defined algebraic varieties via polynomial congruences.

Established Nullstellensatz for congruences.

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 $ \sigma $ on $ S $ and congruence $ \rho $ on $ B, $ we introduce the $ \rho-$algebraic varieties $ Z_{\rho }(\sigma )(B)$ in $ B^{n}, $ which are the set of zeros in $ B^{n} $ of the system of polynomial $ \rho-$congruence equations given by $ \sigma . $ When $ \rho $ 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.