constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
Where a (right) ideal in a magma is a subset which “absorbs” elements, in that with and also the product ,
so a (right) anti-ideal is a subset which “repels” elements, in that the only way that and have product is if also (e.g. Kharchenko 1991 p. 190).
Analogously for left- and two-sided (anti-)ideals.
For the case of rings further conditions on the additive operation are imposed (e.g. Troelstra & van Dalen 1988, Def. 3.6 on p 402): a subset is a two-sided anti-ideal of if:
.
See also at anti-subalgebra the example of anti-ideals.
Anne Sjerp Troelstra, Dirk van Dalen, Def. 3.6 on p. 402 of: Constructivism in Mathematics – An introduction, Volume II, Studies in Logic and the Foundations of Mathematics 123: North Holland (1988) [ISBN:9780444703583]
V. K. Kharchenko: Automorphisms and Derivations of Associative Rings, Mathematics and its Applications 69, Springer (1991) [doi:10.1007/978-94-011-3604-4]
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