Algebraic semantics of the \left\{ \rightarrow,\square\right\} -fragment of Propositional Lax Logic

CELANI, SERGIO; DANIELA MONTANGIE

SOFT COMPUTING - (Online)

Elsevier

Año: 2020 vol. 24 p. 813 - 823

1473-7479

In this paper we will study a particular subvariety of Hilbert algebras with amodal operator $square$, called Lax Hilbert algebras. These algebras are thealgebraic semantic of the $left{ square,ightarrowight} $-fragment ofthe Propositional Lax Logic. We shall prove that the set of fixpoints of a LaxHilbert algebra $leftlangle A,squareightangle $ is a Hilbert algebrasuch that its dual space is homeomorphic to the subspace of reflexive elementsof the corresponding dual space of $A$. We will define the notion of subframeof a Hilbert space $leftlangle X,mathcal{K}ightangle $, and we willprove that there is a 1-1 correspondence between subframes of $leftlangleX,mathcal{K}ightangle $ and binary relations $Qsubseteq Ximes X$ suchthat $leftlangle X,mathcal{K},Qightangle $ is a Lax Hilbert space. Inaddition, we will define the notion of subframe variety and we will prove thatany variety of Hilbert algebras is a subframe variety.