Search results
Results From The WOW.Com Content Network
Hasse diagram of a complemented lattice. A point p and a line l of the Fano plane are complements if and only if p does not lie on l.. In the mathematical discipline of order theory, 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 satisfying a ∨ b = 1 and a ∧ b = 0.
An orthocomplemented lattice is complemented. (def) 8. 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 ...
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).
The lattice L itself is called a pseudocomplemented lattice if every element of L is pseudocomplemented. Every pseudocomplemented lattice is necessarily bounded , i.e. it has a 1 as well. Since the pseudocomplement is unique by definition (if it exists), a pseudocomplemented lattice can be endowed with a unary operation * mapping every element ...
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.
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 is the least element. Modular lattice: a lattice whose elements satisfy the additional modular identity.
Every interval of a geometric lattice (the subset of the lattice between given lower and upper bound elements) is itself geometric; taking an interval of a geometric lattice corresponds to forming a minor of the associated matroid. Geometric lattices are complemented, and because of the interval property they are also relatively complemented. [7]
Groups whose lattice of subgroups is a complemented lattice are called complemented groups (Zacher 1953), and groups whose lattice of subgroups are modular lattices are called Iwasawa groups or modular groups (Iwasawa 1941). Lattice-theoretic characterizations of this type also exist for solvable groups and perfect groups (Suzuki 1951).