Semilattice
Semilattice In mathematical order theory, a semilattice is a partially ordered set (poset) within ... either all binary sets have a supremum (join) or all binary sets have an infimum ... meet). Consequently, one speaks of either a join-semilattice or a meet-semilattice. Semilattices provide ...
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Talk:Semilattice
Talk:Semilattice Please forgive me for asking a question ... following statement in the article on the Semilattice: "Using an easy induction argument, one can ... of non-empty finite subsets in any join-semilattice." Couldn't this be rephrased as: "By ... any non-empty finite subset within any join-semilattice." I realize this is a ...
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Complete lattice
... which all subsets have both a supremum (join) and an infimum (meet). Complete lattices appear ... meet) and a least upper bound (supremum, join). These are denoted by: $\backslash bigwedge$\bigveeA (join). Note that in the special case where ... the greatest element of L. Likewise, the join of the empty set yields the least ... some authors use the terms complete meet-semilattice or complete join-semilattice as another ...
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Glossary of order theory
... directed complete poset with least element. Complete semilattice. The notion of a complete semilattice is defined in different ways. As explained ... lattice. Hence the notion of a complete semilattice is sometimes used to coincide with the ... equivalent to its order dual. A meet-semilattice is distributive if for all elements a ... X such that x R x. J Join. See supremum. L Lattice. A lattice ...
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Lattice (order)
... nonempty finite subsets have both a supremum (join) and an infimum (meet). Lattices can also ... y} has both a least upper bound (join, or supremum) and a greatest lower bound ... meet, or infimum). In this situation, the join and meet of x and y are ... to be both a meet- and a join-semilattice. A lattice with a greatest element ...
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Distributivity (order theory)
... and infima provide the total operations of join ($\backslash vee$) and meet ($\backslash wedge\; ...\; to\; the\; concepts\; of\; meet-semilattices\; and$join-semilattices, respectively. Obviously, in the presence of ... to the interaction of the meet (resp. join) operation with the given order, one can ... a reasonable notion of distributivity. A meet-semilattice is distributive, if if a $\backslash wedge\; ...\; all\; elements\; a,\; b,\; and\; x.\; Distributive$join-semilattices are defined dually. This definition ...
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Completeness (order theory)
... context that the terms meet for ^ and join for v are most common. A poset ... known to exist is therefore called a join-semilattice. The dual notion is meet-semilattice. Further completeness conditions The strongest form of ... iff it is a cpo and a join-semilattice. Indeed, for any subset X, ...
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Compact element
... If the poset P additionally is a join-semilattice (i.e. if it has binary suprema ... easily verified. For the case of the join-semilattices note that any set can be ... course be dropped. Note also that a join-semilattice which is directed complete is almost ...
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Interior algebra
... under arbitrary joins (i.e. if a join of an arbitrary subset of T exists ... For every element b of B, the join ∑{a ∈T : a ≤ b} exists ... i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra. Modal logic Given ...
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List of order topics
... element Upper bound Least upper bound (supremum, join) Greatest lower bound (infimum, meet) Limit superior ... set Dense set Meet-dense set and join-dense set Directed set Net (mathematics) Upper ... Ordinal number Well-quasi-ordering Completeness properties Semilattice Lattice (Directed) complete partial order, (d)cpo ...
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