Locally finite variety
In universal algebra, a variety of algebras means the class of all algebraic structures of a given signature satisfying a given set of identities. One calls a variety locally finite if every finitely generated algebra has finite cardinality, or equivalently, if every finitely generated free algebra has finite cardinality.
The variety of Boolean algebras constitutes a famous example. The free Boolean algebra on n generators has cardinality 22n, consisting of the n-ary operations 2n→2.
The variety of sets constitutes a degenerate example: the free set on n generators has cardinality n, consisting of just the generators themselves.
The variety of pointed sets constitutes a trivial example: the free pointed set on n generators has cardinality n+1, consisting of the generators along with the basepoint.
The variety of graphs defined as follows constitutes a combinatorial example. Define a graph G = (E,s,t) to be a set E of edges and unary operations s, t of source and target satisfying s(s(e)) = t(s(e)) and s(t(e)) = t(t(e)). Vertices are those edges in the (common) image of s and t. The free graph on n generators has cardinality 3n and consists of n edges e each with two endpoints s(e) and t(e). Graphs with nontrivial incidence relations arise as quotients of free graphs, most usefully by identifying vertices.
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