symmetric monoidal (∞,1)-category of spectra
In material set theory, an anti-ideal of a commutative ring $R$ with an apartness relation $\#$ is an $\#$-open subset $I \subseteq R$ which satisfies
In dependently sorted set theory, where membership $x \in S$ is not a relation, the above statement that $x \in I$ for every element $x \in R$ in material set theory is equivalently a predicate in the logic $x \in R \vdash P_I(x)$. The given anti-ideal $I$ is then defined by restricted separation as the set $I \coloneqq \{x \in R \vert P_I(x)\}$, which in structural set theory automatically comes with an injection
such that
Hence, the notion of anti-ideal predicate, a formulation of the notion of anti-ideal as a predicate rather than a subset.
Given a commutative ring $R$ with an apartness relation $\#$, an anti-ideal predicate on $R$ is an $\#$-open predicate $x \in R \vdash P_I(x)$ which satisfies
The anti-ideal $I$ is then defined by restricted separation as $I \coloneqq \{x \in R \vert P_I(x)\}$
The various definitions of anti-ideals translate over from material set theory to anti-ideal predicates in dependently sorted structural set theory by replacing $x \in I$ with $P_I(x)$ throughout the definition:
Created on January 13, 2023 at 00:02:26. See the history of this page for a list of all contributions to it.