subsets with cardinality strictly less than is sometimes denoted or . Similarly, the set of non-empty subsets of might be denoted by or . Power object A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective, the idea of the power set of as the set of subsets of generalizes naturally to the subalgebras of an algebraic structure or algebra. The power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard, subalgebras behave analogously to subsets. However, there are two important properties of subsets that do not carry over to subalgebras in general. First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice. Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set {0,1} = 2, there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way. Certain classes of algebras enjoy both of these properties. The first property is more common, the case of having both is relatively rare. One class that does have both is that of multigraphs. Given two multigraphs and , a homomorphism consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set of homomorphisms from to can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore, the subgraphs of a multigraph are in bijection with the graph homomorphisms from to the multigraph definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of as the multigraph