Lattices of subalgebras

Certain algebraic structures like semilattices have partial orders naturally defined on their elements, while a great many others like groups do not. In fact, groups do not have any natural ordering established on their elements at all. This means that to relate order theory to group theory, a different setting is required other then the elements. This is provided by sets, which are naturally partial ordered by inclusion. The condition of being closed under the operations of the algebraic structure provides for a set of sets of elements which forms a lattice called Sub(A).

Definition. let A be an algebraic structure with a set of n-ary operations defined on A then the set of subsets of the ground set of A that are closed under all the operations of A forms a lattice of sets called Sub(A).

The set of n-ary operations on A is the primary determinant of what Sub(A) will be. Groups have signature (0,1,2) consisting of an identity, inverse, and the group operation and semigroups have signature (2) consisting only of the semigroup operation. Groups as semigroups have strictly more operations but then semigroups but they have fewer subalgebras. This leads to the first deduction of universal algebra, which is that the set of operations of A is inversely proportional to the size of Sub(A).

The torsion-free additive group $(\mathbb{Z},+)$ can take at least three forms depending upon the signature. It can simply a group, a group-as-a-monoid, or a group-as-a-semigroup. Even though they are all defined by the same operation, it is ontologically important to distinguish between them in order to determine Sub(A). The positive integers $\mathbb{Z}+$ form a subsemigroup, the non-negative integers $\mathbb{N}$ form a submonoid, and the even integers $2\mathbb{Z}$ form a subgroup. In this case, the lattice of subsemigroups has strictly more elements then the lattice of submonoids, which in turn has strictly more elements then the lattice of subgroups.

It is not enough to consider the lattice of subalgebras Sub(A). When considering a partial order, one always needs to consider suborders. A set of subalgebras $S \subseteq A$ forms a subalgebra system. Chains of subalgebras are one type of subalgebra system. For example, solvability in groups is determined by a bounds-maintaing subnormal quotient-abelian chain of subgroups. Radical extensions in fields are determined by chains of field extensions such that consecutive filed extensions are formed by simple radicals. This leads to the definition of a subalgebra system.

Definition. let A be an algebraic structure. Then a set $S \subseteq Sub(A)$ of subalgebras of A is a subalgebra system.

Subalgebra systems are also set systems. Suppose, that S is a finite semigroup, then the set of maximal subgroups of S forms a pairwise disjoint sperner family by Green's theorem. The maximal proper subsemigroups of a finite semigroup form a sperner family, which is not necessarily disjoint. Sub(A) itself forms a Moore family as a set system. The closure operation associated with the Moore family can be used to define other sets that are not necessarily subalgebras like minimal generating sets, in the same manner typically done in set theory. Examples of subalgebra systems abound in abstract algebra.