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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
A lower bound of a subset X of a poset P is an element b of P, such that b ≤ x, for all x in X. The dual notion is called upper bound. Lower set. A subset X of a poset P is called a lower set if, for all elements x in X and p in P, p ≤ x implies that p is contained in X. The dual notion is called upper set.
Complete lattice: a lattice in which arbitrary meet and joins exist. Bounded lattice: a lattice with a greatest element and least element. Complemented lattice: a bounded lattice with a unary operation, complementation, denoted by postfix ⊥. The join of an element with its complement is the greatest element, and the meet of the two elements ...
Complete lattice: a lattice in which arbitrary meet and joins exist. Bounded lattice: a lattice with a greatest element and least element. Distributive lattice: a lattice in which each of meet and join distributes over the other. A power set under union and intersection forms a distributive lattice. Boolean algebra: a complemented distributive ...
A complemented lattice is bounded. (def) 9. An algebraic lattice is complete. (def) 10. A complete lattice is bounded. 11. A heyting algebra is bounded. (def) 12. A bounded lattice is a lattice. (def) 13. A heyting algebra is residuated. 14. A residuated lattice is a lattice. (def) 15. A distributive lattice is modular. [3] 16.
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
The set of all ideals of a given semiring form a semiring under addition and multiplication of ideals. Any bounded, distributive lattice is a commutative, semiring under join and meet. A Boolean algebra is a special case of these. A Boolean ring is also a semiring (indeed, a ring) but it is not idempotent under addition.
By an argument similar to the above, one finds that the supremum of a set with upper bounds is the infimum of the set of upper bounds. Consequently, bounded completeness is equivalent to the existence of all non-empty infima. A poset is a complete lattice if and only if it is a cpo and a join-semilattice.