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In a partially ordered set ''P'', the join and meet of a subset ''S'' are respectively the supremum (least upper bound) of ''S'', denoted ⋁''S'', and infimum (greatest lower bound) of ''S'', denoted ⋀''S''. In general, the join and meet of a subset of a partially ordered set need not exist; when they do exist, they are elements of ''P''. Join and meet can also be defined as a commutative, associative and idempotent partial binary operation on pairs of elements from ''P''. If a and b are elements from ''P'', the join is denoted as a ∨ b and the meet is denoted a ∧ b. Join and meet are symmetric duals with respect to order inversion. The join/meet of a subset of a totally ordered set is simply its maximal/minimal element. A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms. ==Partial order approach== Let ''A'' be a set with a partial order ≤, and let ''x'' and ''y'' be two elements in ''A''. An element ''z'' of ''A'' is the meet (or greatest lower bound or infimum) of ''x'' and ''y'', if the following two conditions are satisfied: # ''z'' ≤ ''x'' and ''z'' ≤ ''y'' (i.e., ''z'' is a lower bound of ''x'' and ''y''). # For any ''w'' in ''A'', such that and , we have (i.e., ''z'' is greater than or equal to any other lower bound of ''x'' and ''y''). If there is a meet of ''x'' and ''y'', then it is unique, since if both ''z'' and ''z''′ are greatest lower bounds of ''x'' and ''y'', then and , and thus ''z'' = ''z''′. If the meet does exist, it is denoted . Some pairs of elements in ''A'' may lack a meet, either since they have no lower bound at all, or since none of their lower bounds is greater than all the others. If all pairs of elements have meets, then the meet is a binary operation on ''A'', and it is easy to see that this operation fulfills the following three conditions: For any elements ''x'', ''y'', and ''z'' in ''A'', :a. ''x'' ∧ ''y'' = ''y'' ∧ ''x'' (commutativity), :b. ''x'' ∧ (''y'' ∧ ''z'') = (''x'' ∧ ''y'') ∧ ''z'' (associativity), and :c. ''x'' ∧ ''x'' = ''x'' (idempotency). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「join and meet」の詳細全文を読む スポンサード リンク
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