$Z$-ideals and $Z$-congruences on semiring $\mathcal{R}^+(L)$

Articles in Press

Document Type : Research Paper

Authors

1 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

2 Department of Basic Sciences, Birjand University of Technology, Birjand, Iran

10.48308/cgasa.2024.235948.1496

Abstract

 For a frame $L$, $\mathcal{R}^+(L)$ denotes the nonnegative real valued continuous functions on $L$. We define the concept of $z$-ideals in this  semiring and  give a characterization of  its   $z$-ideals in terms of cozero elements of $L$. Also, we show that there is a one-one correspondence between  $z$-ideals and $z$-congruences on a ring $\mathcal{R}(L)$ and a semiring $\mathcal{R}^+(L)$.  We establish a relationship between $z$-congruence relation on $\mathcal{R}(L)$ and $z$-congruence relation on $\mathcal{R}^+(L)$.  A new characterization of $P$-frames is given via    $z$-congruences on $\mathcal{R}^+(L)$. Also, we show that there is a bijection between the minimal prime ideals  of $\mathcal{R}(L)$ and  coz-ultrafilter on $L$.

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References

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Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 27 December 2024

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