Our work proposes to study the conuclear image of a substructural logic and in particular to investigate the relationship between a substructural logic and its conuclear image. We analyze some axioms familiar to substructural logics and we check if they: (a) are preserved under conuclear images, (b) never hold in a conuclear image, or (c) are compatible with conuclear images but are not necessarily preserved under conuclear images. Moreover, we prove that the conuclear image of any substructural logic has the disjunction property. We finally give a sufficient condition in order that an inequality is preserved under conuclear images and observe that if we slightly relax this condition, we meet counterexamples of inequalities that are not preserved under conuclear images.
We investigate the algebras for the double-power monad on the Sierpisnki space in the category Equ of equilogical spaces, a cartesian closed extension of Top 0 introduced by Scott, and the relationship of such algebras with frames. In particular, we focus our attention on interesting subcategories of Equ. We prove uniqueness of the algebraic structure for a large class of equilogical spaces, and we characterize the algebras for the double-power monad in the category of algebraic lattices and in the category of continuous lattices, seen as full subcategories of Equ. We also analyse the case of algebras in the category Top 0 of T0-spaces, again seen as a full subcategoy of Equ, proving that each algebra for the double-power monad in Top 0 has an underlying sober, compact, connected space.
We study the algebras for the double power monad on the Sierpiński space in the Cartesian closed category of equilogical spaces and produce a connection of the algebras with frames. The results hint at a possible synthetic, constructive approach to frames via algebras, in line with that considered in Abstract Stone Duality by Paul Taylor and others.
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