(0,1)-category theory: logic, order theory

(0,1)-category

proset, partially ordered set (directed set, total order, linear order)

top, true,

bottom, false

implication

filter, interval

lattice, semilattice

meet, logical conjunction, and

join, logical disjunction, or

compact element

lattice of subobjects

complete lattice, algebraic lattice

distributive lattice, completely distributive lattice, canonical extension

hyperdoctrine

(0,1)-topos

Heyting algebra

regular element

Boolean algebra

frame, locale

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A kind of modular lattice.