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Explicitly, a Boolean algebra is the partial order on subsets defined by inclusion (Skiena 1990, p. 207), i.e., the Boolean algebra of a set is the set of subsets of that can be obtained by means of a finite number of the set operations union (OR), intersection (AND), and complementation (NOT) (Comtet 1974, p. 185).
Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky (2002, 2003, 2007). A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose union generates the algebra A and which satisfy various conditions.luster algebras of rank 2 [edit]. Suppose that we start with the cluster { x 1, x 2 } and take the exchange matrix with b 12 =âb 21 =1. Then mutation gives a sequence of variables x 1, x 2, x 3, x 4 ,... such that the clusters are given by adjacent pairs { x n, x n+1 }.