Abstract
Complemented subsets were introduced by Bishop, in order to avoid complementation in terms of negation. In his two approaches to measure theory Bishop used two sets of operations on complemented subsets. Here we study these two algebras and we introduce the notion of Bishop algebra as an abstraction of their common structure. We translate constructively the classical bijection between subsets and boolean-valued functions by establishing a bijection between the proper classes of complemented subsets and of strongly extensional, boolean-valued, partial functions. Avoiding negatively defined concepts, most of our results are within minimal logic.
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- 1.
The type-theoretic interpretation of Bishop’s set theory into the theory of setoids [17, 18] is nowadays the standard way to understand Bishop sets. A categorical interpretation of Bishop sets is Palmgren’s constructive adaptation [16] of Lawvere’s elementary theory of the category of sets. In [9] Coquand views Bishop sets as a natural sub-presheaf of the universe in the cubical set model.
- 2.
If X, Y are totalities an assignment routine f from X to Y is denoted by \(f :X \rightsquigarrow Y\). If X, Y are sets, \(f :X \rightsquigarrow Y\) is a function, if it respects their equalities. Their set is denoted by \(\mathbb {F}(X, Y)\). A function is an embedding, if it is an injection.
- 3.
For the exact definition of this concept within \(\mathrm {BST}\), see [20], Sect. 4.9.
- 4.
The dual to condition (ii), which is equivalent to it, is the equality
$$ (\boldsymbol{A} \cap - \boldsymbol{A}) \cup \bigg (\boldsymbol{A} \cap \bigcup _{i \in I} \boldsymbol{\lambda }(i)\bigg ) =_{{\mathcal {P}^{\boldsymbol{]} \boldsymbol{[}}(X)}} (\boldsymbol{A} \cap - \boldsymbol{A}) \cup \bigg [\bigcup _{i \in I} \big (\boldsymbol{A} \cap \boldsymbol{\lambda }(i)\big )\bigg ],$$and it is the constructive counterpart to \(\boldsymbol{A} \cap \bigcup _{i \in I} \boldsymbol{\lambda }(i) =_{{\mathcal {P}^{\boldsymbol{]} \boldsymbol{[}}(X)}} \bigcup _{i \in I} \big (\boldsymbol{A} \cap \boldsymbol{\lambda }(i)\big )\).
- 5.
These definitions are also used in [8], in order to show that the second algebra of complemented subsets is a Boolean semiring with unit.
- 6.
In [4], p. 74, Bishop and Bridges mention that \(\mathfrak {B}_2(X)\) satisfies “all the usual finite algebraic laws that do not involve the operation of set complementation”. In [8], p. 695, Coquand and Palmgren rightly notice that \(\mathfrak {B}_2(X)\) does not satisfy the absorption equalities \((\boldsymbol{A} \wedge \boldsymbol{B}) \vee \boldsymbol{A} = \boldsymbol{A}\) and \((\boldsymbol{A} \vee \boldsymbol{B}) \wedge \boldsymbol{A} = \boldsymbol{A}\).
- 7.
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Petrakis, I., Wessel, D. (2022). Algebras of Complemented Subsets. In: Berger, U., Franklin, J.N.Y., Manea, F., Pauly, A. (eds) Revolutions and Revelations in Computability. CiE 2022. Lecture Notes in Computer Science, vol 13359. Springer, Cham. https://doi.org/10.1007/978-3-031-08740-0_21
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