Algebraic Definition
A "meet-semilattice" is an algebraic structure 〈S, ∧〉 consisting of a set S with a binary operation ∧, called meet, such that for all members x, y, and z of S, the following identities hold:
- Associativity
- x ∧ (y ∧ z) = (x ∧ y) ∧ z
- Commutativity
- x ∧ y = y ∧ x
- Idempotency
- x ∧ x = x
A meet-semilattice 〈S, ∧〉 is bounded if S includes an identity element 1 such that x ∧ 1 = x for all x in S.
If the symbol ∨, called join, replaces ∧ in the definition just given, the structure is called a join-semilattice. One can be ambivalent about the particular choice of symbol for the operation, and speak simply of semilattices.
A semilattice is an idempotent, commutative semigroup. Alternatively, a semilattice is a commutative band. A bounded semilattice is an idempotent commutative monoid.
A partial order is induced on a meet-semilattice by setting x≤y whenever x∧y=x. For a join-semilattice, the order is induced by setting x≤y whenever x∨y=y. In a bounded meet-semilattice, the identity 1 is the greatest element of S. Similarly, an identity element in a join semilattice is a least element.
Read more about this topic: Semilattice
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