* Complemented lattice

# REDIRECT Complemented lattice

# REDIRECT Complemented lattice

Complemented with similar astronomical instruments, the two sites allow US astronomers to make observations over the entire sky.

Complemented with servants the overall number of the Russian army could be 13, 000-15, 000 men.

Complemented by the refined MacDowell, his gracious, generous performance is never dominating, and their exchanges offer unexpected pleasures.

Complemented by Thai government programs to build minor roads, these projects did much to bind rural Isan more closely to the cities and to Bangkok.

Complemented by a nice shopping street and lots of restaurants, pubs, and nightclubs, Margate is a spot where lots of South Africans and foreign visitors spend their vacation.

Complemented by the work of Justices Kitto, Fullagar and Windeyer, Dixon led what NSW Chief Justice Jim Spigelman has described as " one of the great common law benches of history.

* Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix " ' ", and governed by the axioms x ' x = 1, x ( x ' y )

* A bounded distributive lattice with an involution satisfying De Morgan's laws ( i. e. a De Morgan algebra ), additionally satisfying the inequality x ∧− x ≤ y ∨− y.

* A Heyting algebra is a Cartesian closed ( bounded ) lattice.

In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice ( with join and meet operations written ∧ and ∨ and with least element 0 and greatest element 1 ) equipped with a binary operation a → b of implication such that ( a → b )∧ a ≤ b, and moreover a → b is the greatest such in the sense that if c ∧ a ≤ b then c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound.

In the finite case every nonempty distributive lattice, in particular every nonempty finite chain, is automatically bounded and complete and hence a Heyting algebra.

A Heyting algebra is a bounded lattice such that for all and in there is a greatest element of such that

A bounded lattice is a Heyting algebra if and only if all mappings are the lower adjoint of a monotone Galois connection.

Given a bounded lattice with largest and smallest elements 1 and 0, and a binary operation, these together form a Heyting algebra if and only if the following hold:

* Every totally ordered set that is a bounded lattice is also a Heyting algebra, where is equal to when, and 1 otherwise.

* Any bounded, distributive lattice is a commutative, idempotent semiring under join and meet.

Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element.

Indeed, they are formally defined as the algebra of bounded operators leaving invariant each subspace contained in a subspace nest, that is, a set of subspaces which is totally ordered by inclusion and is also a complete lattice.

# X is homeomorphic to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic ( as a bounded lattice ) to the lattice K < sup ></ sup >( X ) ( this is called Stone representation of distributive lattices ).

Likewise, " bounded complete lattice " is almost unambiguous, since one would not state the boundedness property for complete lattices, where it is implied anyway.

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 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

-- The crystallization of copolymers comprising Af units interspersed with a minor percentage of Af is limited by the inability of the crystal lattice characteristic of the former to accommodate the bulky side group of the latter.

Although the neutral ground-state chemical compounds of argon are presently limited to HArF, argon can form clathrates with water when atoms of it are trapped in a lattice of the water molecules.

The crystal lattice of solid americium and its compounds contains intrinsic defects, which are induced by self-irradiation with alpha particles and accumulate with time ; this results in a drift of some material properties.

Amethyst is a purple variety of quartz ( SiO < sub > 2 </ sub >) and owes its violet color to irradiation, iron impurities ( in some cases in conjunction with transition element impurities ), and the presence of trace elements, which result in complex crystal lattice substitutions.

He ordained further that some should be called " Abbreviators of the Upper Bar " ( Abbreviatores de Parco Majori ; the name derived from a space in the chancery, surrounded by a grating, in which the officials sat, which is called higher or lower ( major or minor ) according to the proximity of the seats to that of the vice-chancellor ), the others of the Lower Bar ( Abbreviatores de Parco Minori ); that the former should sit upon a slightly raised portion of the chamber, separated from the rest of the hall or chamber by lattice work, assist the Cardinal Vice-Chancellor, subscribe the letters and have the principal part in examining, revising, and expediting the apostolic letters to be issued with the leaden seal ; that the latter, however, should sit among the apostolic writers upon benches in the lower part of the chamber, and their duty was to carry the signed schedules or supplications to the prelates of the upper bar.

Upon heating, α-berkelium transforms into another phase with an fcc lattice ( but slightly different from β-berkelium ), space group Fmm and the lattice constant of 500 pm ; this fcc structure is equivalent to the closest packing with the sequence ABC.

This deformation of the lattice causes another electron, with opposite spin, to move into the region of higher positive charge density.

Half or more of the trench is filled with " green " ( unfired ) bricks which are stacked in an open lattice pattern to allow airflow.

The lattice is capped with a roofing layer of finished brick.

Also called the " caesium chloride structure ", this structural motif is composed of a primitive cubic lattice with a two-atom basis, each with an eightfold coordination ; the chloride atoms lie upon the lattice points at the edges of the cube, while the caesium atoms lie in the holes in the center of the cubes.

Since diamond also consists of carbon arranged in a crystalline lattice ( though of a different configuration ), scientists have nicknamed this star " Lucy " after the Beatles song " Lucy in the Sky with Diamonds.

A successful mathematical classification method for physical lattice defects, which works not only with the theory of dislocations and other defects in crystals but also, e. g., for disclinations in liquid crystals and for excitations in superfluid < sup > 3 </ sup > He, is the topological homotopy theory.

The high radioactivity of einsteinium-253 produces a visible glow and rapidly damages its crystalline metal lattice, with released heat of about 1000 watts per gram.

* It is useful to define gcd ( 0, 0 ) = 0 and lcm ( 0, 0 ) = 0 because then the natural numbers become a complete distributive lattice with gcd as meet and lcm as join operation.

UV light creates electron-hole pairs, with the holes reacting with lattice oxygen, creating surface oxygen vacancies, while the electrons reduce V < sup > 5 +</ sup > to V < sup > 3 +</ sup >.

Vacancies are crystal lattice points unoccupied by an atom: in this case the ion collides with a target atom, resulting in transfer of a significant amount of energy to the target atom such that it leaves its crystal site.

This deformation of the lattice causes another electron, with opposite " spin ", to move into the region of higher positive charge density.