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In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") between two bounded lattices and should also have the following property: =, =. In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins.
A complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element b such that a ∨ b = 1 and a ∧ b = 0. In general an element may have more than one complement. However, in a (bounded) distributive lattice every element will have at most one complement. [1]
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.
Likewise, "bounded complete lattice" is almost unambiguous, since one would not state the boundedness property for complete lattices, where it is implied anyway. Also note that the empty set usually has upper bounds (if the poset is non-empty) and thus a bounded-complete poset has a least element.
A join-semilattice is bounded if it has a least element, the join of the empty set. Dually, a meet-semilattice is bounded if it has a greatest element, the meet of the empty set. Other properties may be assumed; see the article on completeness in order theory for more discussion on this subject.
An alternative way of stating the same fact is that every distributive lattice is a subdirect product of copies of the two-element chain, or that the only subdirectly irreducible member of the class of distributive lattices is the two-element chain. As a corollary, every Boolean lattice has this property as well. [6]
A bounded lattice H is a Heyting algebra if and only if every mapping f a is the lower adjoint of a monotone Galois connection. In this case the respective upper adjoint g a is given by g a ( x ) = a → x , where → is defined as above.
A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete.