nLab
Boolean prealgebra object

Contents

Context

Relations

Category theory

Limits and colimits

(0,1)(0,1)-Category theory

Contents

Idea

A version of a Boolean algebra without the preorder being an antisymmetric relation, internal to a finitely complete category.

Definition

In a finitely complete category CC, a Boolean prealgebra object is a bicartesian closed preordered object XX with a function

ζ:((*X)×(*X))(*R)\zeta:((* \to X) \times (* \to X)) \to (* \to R)

such that for all global elements a:*Xa:* \to X and b:*Xb:* \to X,

  • sζ(a,b)=abs \circ \zeta(a, b) = a \Rightarrow b
  • tζ(a,b)=(a)bt \circ \zeta(a, b) = (a \Rightarrow \bot) \vee b

See also

Last revised on May 14, 2022 at 10:41:38. See the history of this page for a list of all contributions to it.