$L$-algebraic system and its reflectivity

Year 2026, Volume: 55 Issue: 1 , 162 - 175 , 23.02.2026

Mengying Liu , Yueli Yue

https://doi.org/10.15672/hujms.1697171

https://izlik.org/JA82YA92TE

Abstract

In this paper, with a continuous lattice $L$ as the truth valued table, we first prove that the non-topological category $L$-${\bf AlgSys}$ of $L$-algebraic systems can be embedded into the topological category of variety-based $(A,L)$-fuzzy algebraic closure spaces. Subsequently, we demonstrate that the Sierpinski $L$-algebraic system $(L,\mathcal{S},{\models}_{\mathcal{S}})$ is an injective object in the category $L$-${\bf AlgSys}_0$ of $S_0$-$L$-algebraic systems. Furthermore, we prove that $L$-${\bf AlgSys}_0$ is epireflective in $L$-${\bf AlgSys}$, while the category $L$-${\bf SobAlgSys}$ of sober $L$-algebraic systems is reflective in $L$-${\bf AlgSys}$. Finally, we consider the relationships between the category of $L$-algebraic closure spaces and that of strong $L$-algebraic systems, and between the category of continuous lattices and that of sober $L$-algebraic systems.

Keywords

L-algebraic system , Sierpinski object , sober , reflective , L-algebraic closure space

Supporting Institution

the National Natural Science Foundation of China

Project Number

12371467

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