In constructive mathematics, we often do algebra by equipping an algebra with a tight apartness (and requiring the algebraic operations to be strongly extensional). In this context, it is convenient to replace subalgebras with anti-subalgebras, which classically are simply the complements of subalgebras.
Let us work in the context of universal algebra, so an algebra is a set equipped with a family of functions (where each arity? is a cardinal number) that satisfy certain equational identities (which are irrelevant here). As usual, a subalgebra of is a subset such that whenever each .
Now we require to have a tight apartness , which induces a tight apartness on each (via existential quantification), and we require the operations to be strongly extensional. (There is no need to require that any arity be finite or that there be finitely many .)
A subset of is open (relative to ) if or whenever . An antisubalgebra of is an open subset such that some whenever . By taking the contrapositive?, we see that the complement of is a subalgebra ; then may be recovered as the -complement of (the set of those such that whenever ). However, we cannot start with an arbitrary subalgebra and get an antisubalgebra in this way, as we cannot (in general) prove openness. (We can take the antisubalgebra generated by the -complement of , as described below, but its complement will generally only be a superset of .)
The empty subset of any algebra is an antisubalgebra, the empty antisubalgebra or improper antisubalgebra, whose complement is the improper subalgebra (which is all of ). An antisubalgebra is proper if it is inhabited; the ability to have a positive definition of when an antisubalgebra is proper is a significant motivation for the concept.
If is an antisubalgebra and is a constant (given by an operation or a composite of same with other operations), then whenever . If there are only Kuratowski-finitely many constants (which is needed to prove openness), we define the trivial antisubalgebra to be the subset of those elements such that for each constant (the -complement of the trivial subalgebra?). In general, we may also take the trivial antisubalgebra to be the union of all antisubalgebras, although this is not predicative.
Instead of subgroups, use antisubgroups. In detail, is an antisubgroup if whenever , or whenever , and whenever . An antisubgroup is normal if whenever . The trivial antisubgroup is the -complement of .
Instead of ideals (of commutative rings), use antiideals (and we also have left and right antiideals of general rings). In detail, is an antiideal if whenever , or whenever , and whenever . It follows that an antiideal is proper iff . is prime if it is proper and whenever and ; is minimal if it is proper and, for each , for some , for each , (which is constructively stronger than being prime and minimal among proper ideals). The trivial? antiideal is the -complement of .