In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

* Boolean algebra: a complemented distributive lattice.

By virtue of J1a, it is a complemented lattice whose upper bound is ().

* A complemented distributive lattice.

In the mathematical discipline of order theory, a complemented lattice is a bounded lattice in which every element a has a complement, i. e. an element b satisfying a ∨ b = 1 and a ∧ b = 0.

A relatively complemented lattice is a lattice such that every interval is complemented.

An orthocomplementation on a complemented lattice is an involution which is order-reversing and maps each element to a complement.

Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra.

A lattice in which every element has exactly one complement is called a uniquely complemented lattice.

A lattice with the property that every interval is complemented is called a relatively complemented lattice.

In other words, a relatively complemented lattice is characterized by the property that for every element a in an interval d there is an element b such that

A distributive lattice is complemented if and only if it is bounded and relatively complemented.

Image: Hexagon lattice. svg | The hexagon lattice admits a unique orthocomplementation, but it is not uniquely complemented.

The design is complemented by joinery windows set with multicolored stained glass.

The CPC models ' hardware is based on the Zilog Z80A CPU, complemented with either 64 or 128 kB of memory.

To Clement, scripture is an innately true primitive philosophy which is complemented by human reason through the Logos.

The European Company Regulation is complemented by an Employee Involvement Directive which sets rules for participation by employees on the company's board of directors.

The regulation is complemented by the Council Directive supplementing the Statute for a European Company with regard to the involvement of employees ( informally " Council Directive on Employee Participation "), adopted 8 October 2001.

The function of an MTA is usually complemented with some means for email clients to access stored messages.

In large land-based electricity generation steam turbines, the shrouding is often complemented, especially in the long blades of a low-pressure turbine, with lacing wires.

The Coptic manuscript itself has been dated to the 4th century, however, it is complemented by a few fragments in Greek dating from the 3rd century, implying an earlier date.

Defenders of traditions and faiths from claims that they are irrationalist for ignoring or even attempting to forbid reason and argument concerning some subjects, typically maintain that there is no real conflict with reason, because reason itself is not enough to explain such things as the origins of the universe, or right and wrong, and so reason can and should be complemented by other sources of knowledge, or in other words first principles.

However, leisure traffic took over, and the canal is now as busy as it ever was, with leisure boating complemented by fishing, towpath walking and gongoozling.

Federalist No. 10, in which Madison discusses the means of preventing rule by majority faction and advocates a large, commercial republic, is generally regarded as the most important of the 85 articles from a philosophical perspective ; it is complemented by Federalist No. 14, in which Madison takes the measure of the United States, declares it appropriate for an extended republic, and concludes with a memorable defense of the constitutional and political creativity of the Federal Convention.

It is one component of a nuclear triad, which is complemented by the Trident submarine-launched ballistic missile ( SLBM ) and by nuclear weapons carried by long-range strategic bombers.

The girls ' uniform is complemented by white ankle socks and black school shoes.

The entrance to the festival is free of charge and beers are sold at low prices, complemented by a mix of international music.

As with cystic echinococcosis, ultrasonography is the imaging technique of choice for alveolar echinococcosis and is usually complemented by CT scans since CT scans are able to detect the largest number of lesions and calcifications that are characteristic of alveolar echinococcosis.

The town's three main trading streets of High Street, West Street and South Street are complemented by a small open-air shopping centre, St Martin's Walk, which is adjacent to the town's main car park and easily accessed from the High Street.

Francesco Sansovino described such a procession in minute detail in 1581, and his verbal description is confirmed and complemented by Cesare Vecellio's 1586 painting of a ducal procession in the Piazza San Marco.

This is generally complemented with the concept of God having transferred the status of " God's people " from the Jews to the Christian Church.

Its modern alphabet is basically a subset of the Latin script ( with " J ", " K " and " Y " but not " W "), complemented with two diacritics and six digraphs.

After its initial release in 1075 / 6, Gesta was complemented with supplementary Scholias until the death of Adam in the 1080s.

These features can be complemented and augmented with links to separate, purpose-built applications for analytics and business intelligence.

A trade agreement with the EEC was complemented by another with the Soviet Bloc.

The battleship design was complemented by the introduction of a variant with lighter armour and greater speed, which became the battlecruiser.

When conquistadores arrived in the Aztec capital Tenochtitlan ( now Mexico City ), they found that the people's diet consisted largely of corn-based dishes with chiles and herbs, usually complemented with beans and tomatoes or nopales.

In 1996, the Pentium with MMX Technology ( often simply referred to as Pentium MMX ) was introduced with the same basic microarchitecture complemented with an MMX instruction set, larger caches, and some other enhancements.

Cinematographer Conrad Hall complemented Mendes ' style with peaceful shot compositions to contrast with the turbulent on-screen events.

After the arrival of the Middle Age newcomers ( see below ), as growing population density necessitated higher crop yields, irrigated rice paddies emerged in Betsileo country by 1600 and were complemented with terraced paddies throughout Imerina a century later.

Eventually, most Nordic countries replaced or complemented this system with the prevailing " international " standard of inherited family names.

Much of this work combines Mackintosh's own designs with those of his wife, whose flowing, floral style complemented his more formal, rectilinear work.

It was also the year Silver City complemented its Gatwick — Le Touquet all-passenger operation with a vehicle ferry service.

Except for certain fertile areas in Hedemarken and Østfold, crops were limited to hardy grains, such as oats, rye, and barley ; and livestock to sheep, goats, cattle, pigs, and some poultry ; in places this was complemented with hunting.