A semilattice is a commutative, idempotent semigroup; i.e., 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.

A semilattice is an algebra S = (S,∗) satisfying, for all x,y,z ∈ S, (1) x∗ x = x, (2) x∗ y = y ∗ x, (3) x∗ (y ∗ z) = (x∗ y)∗ z. In other words, a semilattice is an idempotent commutative semigroup. The symbol ∗ can be replaced by any binary operation symbol, and in fact we will most often

semilattice and a join-semilattice is called a lattice. A poset Lis a meet-semilattice (resp., a join-semilattice) if and only if every two elements of Lhave a meet (resp., a join).

Semilattices include lattices, which in turn include Heyting and Boolean algebras. These lattice-like structures all admit order-theoretic as well as algebraic descriptions. The sub-field of abstract algebra that studies lattices is called lattice theory.

May 19, 2022 · In a join-semilattice the binary join ∨ \vee is commutative, associative, has ⊥ \bot as a unit, and is idempotent: a ∨ a = a a\vee a =a. And in fact, given any commutative and idempotent monoid ( A , ∨ , ⊥ ) (A,\vee,\bot) , we can define a ≤ b a\le b to mean a ∨ b = b a \vee b = b to make it into a poset with finite joins; thus we ...

Given an upper semilattice (|S|, ∨), we shall consider |S| as partially ordered by the unique ordering which makes ∨the least upper bound operation (characterized in two equivalent ways in the above lemma).

A semilattice is a set equipped with a binary operation that is associative, commutative, and idempotent. It can be defined as a poset where the partial ordering is given by X ≤ Y if Y = X Y and every two elements have a supremum.

Nov 23, 2023 · Every semi-lattice $p=\langle p,+\rangle$ can be turned into a partially ordered set (the partial order $\leq$ is defined by the relation $a\leq b$ if and only if $a+b=b$) in which for any pair of elements there is a least upper bound $\sup\ {a,b\}=a+b$.

A semilattice (S, ⋅) (S, ⋅) is a commutative idempotent semigroup. A congruence on a semilattice is an equivalence relation that preserves multiplication, i.e. x1 θy1 x 1 θ y 1 and x2 θ y2 x 2 θ y 2 imply that x1 ⋅x2 θy1 ⋅y2 x 1 ⋅ x 2 θ y 1 ⋅ y 2.

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