In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset...

equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras...

theorem for commutative semigroups in terms of semilattices. A semilattice (or more precisely a meet-semilattice) (L, ≤) is a partially ordered set where every...

arrangement of planes. The intersection semilattice L(A) is a meet semilattice and more specifically is a geometric semilattice. If the arrangement is linear or...

pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that...

In abstract algebra, a branch of mathematics, a maximal semilattice quotient is a commutative monoid derived from another commutative monoid by making...

least lattices, but the concept can in fact reasonably be generalized to semilattices as well. Probably the most common type of distributivity is the one defined...

following semilattice-theoretical formulation of CLP. Semilattice-theoretical formulation of CLP: Is every distributive (∨,0)-semilattice isomorphic...

called a closed sublattice of L. The terms complete meet-semilattice or complete join-semilattice is another way to refer to complete lattices since arbitrary...

valuable alternative presentation. In the case of semilattices, an explicit construction of the free semilattice F ∨ ( X ) {\displaystyle F_{\vee }(X)} is straightforward...

generalized Boolean algebra, while (B, ∨, 0) is a generalized Boolean semilattice. Generalized Boolean lattices are exactly the ideals of Boolean lattices...

cannot be a lattice (or even a meet semilattice), since by definition, every two elements in a lattice (or meet semilattice) must have a common lower bound...

residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras...

semigroup) and idempotents commute (that is, the idempotents of S form a semilattice). Every L {\displaystyle {\mathcal {L}}} -class and every R {\displaystyle...

suprema are known to exist is therefore called a join-semilattice. The dual notion is meet-semilattice. The strongest form of completeness is the existence...

(contrast partially ordered sets, which need not be directed). Join-semilattices (which are partially ordered sets) are directed sets as well, but not...

sup I then c is an element of I. If the poset P additionally is a join-semilattice (i.e., if it has binary suprema) then these conditions are equivalent...

should compute the join for any pair of replica states, and should form a semilattice with the initial state as the neutral element. In particular this means...

relation. Synonym for Connected relation. Complete semilattice. The notion of a complete semilattice is defined in different ways. As explained in the...

independently in the early 1970s by Biryukov, Fennemore and Gerhard. Semilattices, left-zero bands, right-zero bands, rectangular bands, normal bands,...

flag consisting of modular elements. Equivalently, the intersection semilattice of the arrangement is a supersolvable lattice, in the sense of Richard...

y ≤ x. A model of the Plotkin powertheory is a continuous semilattice: it is a semilattice whose carrier is a domain and for which the operation is continuous...

"and"), or equivalently the set {0,1} under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order...

commutative, therefore semilattices (one of them is the three-element totally ordered set, and the other is a three-element semilattice that is not a lattice)...

action algebra is an algebraic structure which is both a residuated semilattice and a Kleene algebra. It adds the star or reflexive transitive closure...

function is a lattice homomorphism, while the image function is only a semilattice homomorphism (that is, it does not always preserve intersections). Bijection...

Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗...

0\cdot x=x} Under these conditions, the operational frame is a join-semilattice. An operational model M {\displaystyle M} is a frame F {\displaystyle...

complemented lattice, i.e. a Boolean algebra. The same holds for any semilattice when "semilattice" is substituted for "distributive lattice" and "subsemilattice"...

Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗...