$L$-algebraic system and its reflectivity

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

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