A ring of sets in the order-theoretic sense forms a
distributive lattice in which the intersection and union operations correspond to the lattice's
meet and join operations, respectively. Conversely, every distributive lattice is isomorphic to a ring of sets; in the case of
finite distributive lattices, this is
Birkhoff's representation theorem and the sets may be taken as the lower sets of a partially ordered set. A family of sets closed under union and relative complement is also closed under
symmetric difference and intersection. Conversely, every family of sets closed under both symmetric difference and intersection is also closed under union and relative complement. This is due to the identities • A \cup B = (A\, \triangle\, B)\, \triangle\, (A \cap B) and • A \setminus B = A\, \triangle\, (A \cap B). Symmetric difference and intersection together give a ring in the measure-theoretic sense the structure of a
boolean ring. In the measure-theoretic sense, a is a ring closed under unions, and a
δ-ring is a ring closed under countable intersections. Explicitly, a σ-ring over X is a set \mathcal{F} such that for any sequence \{A_k\}_{k=1}^\infty \subseteq \mathcal{F}, we have \bigcup_{k=1}^\infty A_k \in \mathcal{F}. Given a set X, a − also called an − is a ring that contains X. This definition entails that an algebra is closed under absolute complement A^c = X \setminus A. A
σ-algebra is an algebra that is also closed under countable unions, or equivalently a σ-ring that contains X. In fact, by
de Morgan's laws, a δ-ring that contains X is necessarily a σ-algebra as well. Fields of sets, and especially σ-algebras, are central to the modern theory of
probability and the definition of
measures. A '''''' is a family of sets \mathcal{S} with the properties \varnothing \in \mathcal{S}, • If (3) holds, then \varnothing \in \mathcal{S} if and only if \mathcal{S} \neq \varnothing. A, B \in \mathcal{S} implies A \cap B \in \mathcal{S}, and A, B \in \mathcal{S} implies A \setminus B = \bigcup_{i=1}^n C_i for some disjoint C_1, \ldots, C_n \in \mathcal{S}. Every ring (in the measure theory sense) is a semi-ring. On the other hand, \mathcal{S} := \{\emptyset,\{x\},\{y\}\} on X = \{x,y\} is a semi-ring but not a ring, since it is not closed under unions. A '
or ' is a collection \mathcal{S} of subsets of X satisfying the semiring properties except with (3) replaced with: • If E \in \mathcal{S} then there exists a finite number of mutually
disjoint sets C_1, \ldots, C_n \in \mathcal{S} such that X \setminus E = \bigcup_{i=1}^n C_i. This condition is stronger than (3), which can be seen as follows. If \mathcal{S} is a semialgebra and E, F \in \mathcal{S}, then we can write F^c = F_1 \cup \ldots \cup F_n for disjoint F_i \in S. Then: E \setminus F = E \cap F^c = E \cap (F_1 \cup \ldots \cup F_n) = (E \cap F_1) \cup \ldots \cup (E \cap F_n) and every E \cap F_i \in S since it is closed under intersection, and disjoint since they are contained in the disjoint F_i's. Moreover the condition is
strictly stronger: any S that is both a ring and a semialgebra is an algebra, hence any ring that is not an algebra is also not a semialgebra (e.g. the collection of finite sets on an infinite set X). ==See also==