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 $X$ equipped with a family of functions $f_i\colon X^{n_i} \to X$ (where each arity $n_i$ is a cardinal number) that satisfy certain equational identities (which are irrelevant here). As usual, a subalgebra of $X$ is a subset $S$ such that $f_i(p_1,\ldots,p_{n_i}) \in S$ whenever each $p_k \in S$. There is no need, in general, to require that any arity $n_i$ be finite or that there be finitely many $f_i$; however, for a few results, we will need a special case of these that we will call having well-behaved constants:
The first item is true of all derived operations in the theory as long as it is true of the fundamental operations in the signature; but in this last item, we're counting all derived constants, not just the fundamental ones. For example, the theory of (unital) rings does not have well-behaved constants, because there are infinitely many constants (one for each integer).
Now we require $S$ to have a tight apartness $\ne$, which induces a tight apartness on each $X^{n_i}$ (via existential quantification), and we require the operations $f_i$ to be strongly extensional. An algebra $X$ with these properties is called an inequality algebra. (For much of the theory we don’t need the apartness to be tight, but for some purposes it is necessary.)
A subset $A$ of $X$ is open (or $\ne$-open) if, whenever $p \in A$, $q \in A$ or $p \ne q$. An antisubalgebra of $X$ is an open subset $A$ such that $p_j \in A$ for some $j$ whenever $f_i(p_1,\ldots,p_{n_i}) \in A$ for any $i$. By taking the contrapositive, we see that the complement of $A$ is a subalgebra $S$, but we cannot (in general) start with a subalgebra $S$ and get an antisubalgebra $A$. (Impredicatively, we can take the antisubalgebra generated, as described below, by the $\ne$-complement of $S$, that is the set of those elements of $X$ that $\ne$ every element of $S$, but its complement will generally only be a superset of $S$.)
Unless otherwise noted, all of the constructions in these examples should be predicative.
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 $X$). 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 $A$ is an antisubalgebra and $c$ is a constant (given by an operation $X^0 \to X$ or a composite of same with other operations), then $p \ne c$ whenever $p \in A$. If the theory has well-behaved constants, then we can define the trivial antisubalgebra to be the subset of those elements $p$ such that $p \ne c$ for each constant $c$ (the $\ne$-complement of the trivial subalgebra). In general, we may also take the trivial antisubalgebra to be the union of all antisubalgebras (but this is not predicative).
Instead of subgroups, use antisubgroups. In this case the definition can be simplified a bit: a subset $A$ of an inequality group $X$ is an antisubgroup if $p \ne 1$ whenever $p \in A$, $p \in A$ or $q \in A$ whenever $p q \in A$, and $p \in A$ whenever $p^{-1} \in A$. We need not assume that $A$ is open; this can be proved from strong extensionality of the group operations on $X$ and the stronger form of the nullary anticlosure condition (“$p \ne 1$ whenever $p \in A$” is a strengthening of the condition $\neg (1\in A)$ that would be the literal nullary case of the general definition.) An antisubgroup $A$ is normal if $p q \in A$ whenever $q p \in A$. The trivial antisubgroup is the $\ne$-complement of $\{1\}$.
Instead of ideals (of rings), use antiideals. (Technically, these are antisubalgebras of the ring as a module over itself.) Again we can omit $\ne$-openness by strengthening the nullary condition. In detail, a subset $A$ of $X$ is a two-sided antiideal (or simply an antiideal in the commutative case) if $p \ne 0$ whenever $p \in A$, $p \in A$ or $q \in A$ whenever $p + q \in A$, and $p \in A$ and $q \in A$ whenever $p q \in A$. $A$ is a left antiideal if instead the last condition requires only that $p \in A$, and $A$ is a right antiideal if instead the last condition requires only that $q \in A$. It follows that an antiideal $A$ is proper iff $1 \in A$. $A$ is prime (or antiprime) if it is proper and $p r q \in A$ for some $r$ whenever $p \in A$ and $q \in A$; in the commutative case, we can say that $p q \in A$ whenever $p \in A$ and $q \in a$. $A$ is minimal (or antimaximal) if it is proper and, for each $p \in A$, for some $q$, for each $r \in A$, $p q + r \ne 1$ and $q p + r \ne 1$ (which is constructively stronger than being prime and minimal among proper ideals); of course, we only need one of these two inequalities in the commutative case. The trivial antiideal is the $\ne$-complement of $\{0\}$.
Note that a union of antisubalgebras is again an antisubalgebra. Given any subset $B$ of $X$, the antisubalgebra generated by $B$ is the union of all antisubalgebras contained in $B$. (This construction is not predicative, although it may still be true predicatively that the generated subalgebra exists in some situations.)
In some cases, we may prefer to anti-generate: given any subset $B$ of $X$, the antisubalgebra antigenerated by $B$ is the union of all antisubalgebras whose elements are all distinct from ($\ne$) each element of $B$, in other words the antisubalgebra generated by the $\ne$-complement of $B$. For example, the trivial antisubgroup of a group is antigenerated by $\{1\}$, and the trivial antiideal of a ring is antigenerated by $\{0\}$. More generally than these examples, we may talk of the cyclic antisubgroup? or principal antiideal antigenerated by a given element of the group or ring.
To form a quotient group or a quotient ring, it's enough to have a normal subgroup or a two-sided ideal. However, if we want the quotient algebra to inherit an apartness from the original algebra, then we need antisubgroups and antiideals.
In general, instead of congruence relations, use anticongruence relations. An anticongruence relation $K$ on $X$ is an apartness relation on $X$ that is also an antisubalgebra of $X \times X$. Given this, let $R$ be the negation of $K$; then $R$ is a congruence relation, giving a quotient algebra $X/R$. Furthermore, $K$ becomes a tight apartness on $X/R$, relative to which the algebra operations on $X/R$ are strongly extensional. We denote the resulting algebra-with-apartness by $X/K$. (This notation should cause no confusion; if an apartness relation on a set $X$ is also an equivalence relation, then $X$ must be the empty set, which has a unique apartness and at most one algebra structure, and the only quotient set of the empty set is itself.) The quotient map $X \twoheadrightarrow X/K$ is also strongly extensional.
Conversely, any strongly extensional map $f\colon X \to Y$ between algebras with apartness gives rise to an anticongruence $\aker f$ on $X$ (the antikernel of $f$), where $(p, q) \in \aker f$ iff $f(p) \ne f(q)$. The complement of the antikernel is (because the apartness of $Y$ is tight) the kernel in the usual sense of universal algebra. Thus, the quotient algebra $X/(\aker f)$ is naturally isomorphic to a subalgebra? $im f$ of $Y$; the maps $X \twoheadrightarrow X/(\aker f) \cong \im f \hookrightarrow Y$ are strongly extensional. Similarly, a sequence $X \overset{f}\to Y \overset{g} \to Z$ is exact iff $\im f$ is the complement of $\aker g$.
(We would like to say that there is an antisubalgebra $\aim f$ of $Y$ whose complement is $\im f$; then we could, for example, define a stronger notion of exactness requiring that $\aker g$ equal the antiimage of $f$. In principle, $\aim f$ should be the $\ne$-complement of $\im f$. If $X$ is Kuratowski-finite, then this works, but in general, we can prove neither that this is open nor that its complement is all of $\im f$.)
Given a group-with-apartness and a normal antisubgroup $A$, we define an anticongruence $K$, where $(p, q) \in K$ iff $p q^{-1} \in A$. Similarly, given a ring-with-apartness and a two-sided antiideal $A$, we define an anticongruence $K$, where $(p, q) \in K$ iff $p - q \in A$. This allows us to form quotient groups or quotient rings by modding out by normal antisubgroups or two-sided antiideals. Conversely, we can interpret the antikernel as a normal antisubgroup or two-sided antiideal: $p \in \aker f$ iff $f(p) \ne 1$, $p \in \aker f$ iff $f(p) \ne 0$, etc. In general, this works for any Omega-group structure.
As noted at apartness relation, an apartness relation on a set $X$ is equivalent to a (strongly) closed equivalence relation on the corresponding discrete locale, and the $\ne$-open subsets are those whose complementary closed sublocales are stable under this equivalence relation, and the $\ne$-topology itself is the corresponding quotient locale. From this point of view, an algebra structure is strongly extensional if it respects the equivalence relation, hence passes to the quotient; and an antisubalgebra is an $\ne$-open set whose complementary closed sublocale is additionally a localic subalgebra, since the operation $\mathsf{C}$ from open sublocales to closed ones takes arbitrary (not only finite) unions to intersections.
In other words, antisubalgebras of an inequality algebra are equivalent to closed subalgebras of a localic algebra, in the case when the latter is the quotient of a discrete algebra by a closed localic congruence.
According to Troelstra and van Dalen:
The study of algebraic structures in an intuitionistic setting was undertaken by Heyting (1941)… in full generality, equipped with an apartness relation. The notion of an antisubstructure, implicit in Heyting’s treatment of ideals in polynomial rings, was formulated explicitly by D.S. Scott (1979) (N.B. the first draft of this paper contains a good deal more than the published version). Ruitenburg (1982, 1982A) deals with intuitionistic algebra in the spirit of Heyting and Scott.
Arend Heyting, Untersuchungen über intuitionistische Algebra, 1941
Dana Scott, Identity and existence in intuitionistic logic, 1979
Wim Ruitenberg?, Intuitionistic Algebra, Ph.D. Thesis, Rijksuniversiteit Utrecht, 1982
Wim Ruitenberg?, Primality and invertibility of polynomials, 1982
Surprisingly, antisubalgebras make hardly any appearence in Ray Mines?, Fred Richman, Wim Ruitenburg?. A Course in Constructive Algebra. Springer, 1987.
More can be found in Anne Troelstra and Dirk van Dalen?, Constructivism in Mathematics (volume 2).
Last revised on August 31, 2018 at 02:31:59. See the history of this page for a list of all contributions to it.