* Complemented lattice: a bounded lattice with a unary operation, complementation, denoted by postfix " ' ".

* 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

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

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

This is true of all components which have such a bounded support.

The tract is bounded by Island Ave., Dicks Ave., 61st St., and Eastwick Ave..

The Atlantic Ocean is bounded on the west by North and South America.

** A uniform space is compact if and only if it is complete and totally bounded.

The region is bounded by the Black Sea to the north, the Mediterranean Sea to the south and the Aegean Sea to the west.

The Anatolian peninsula, also called Asia Minor, is bounded by the Black Sea to the north, the Mediterranean Sea to the south, the Aegean Sea to the west, and the Sea of Marmara to the northwest, which separates Anatolia from Thrace in Europe.

Under this definition, Anatolia is bounded to the East by the Armenian Highland, and the Euphrates before that river bends to the southeast to enter Mesopotamia.

To the southeast, it is bounded by the ranges that separate it from the Orontes valley in Greater Syria and the Mesopotamian plain.

thus, as T decreases, ΔG and ΔH approach each other ( so long as ΔS is bounded ).

We now consider how the choice of description language affects the value of K, and show that the effect of changing the description language is bounded.

The South African plateau, as far as about 12 ° S, is bounded east, west and south by bands of high ground which fall steeply to the coasts.

It lies on Abadan Island ( long, 3 – 19 km or 2 – 12 miles wide, the island is bounded in the west by the Arvand waterway and to the east by the Bahmanshir outlet of the Karun River ), from the Persian Gulf, near the Iraqi-Iran border.

The ACT is bounded by the Goulburn-Cooma railway line in the east, the watershed of Naas Creek in the south, the watershed of the Cotter River in the west, and the watershed of the Molonglo River in the north-east.

The Arabian Sea ( Persian Sea ) is a region of the Indian Ocean bounded on on the north by Pakistan and Iran, on the south by northeastern Somalia, on the east by India and on the west by the Arabian Peninsula.

However, the recursion is bounded because in each recursive application either m decreases, or m remains the same and n decreases.

It is bounded by the Scandinavian Peninsula, the mainland of Europe, and the Danish islands.

The Black Sea is bounded by Europe, Anatolia and the Caucasus and is ultimately connected to the Atlantic Ocean via the Mediterranean and the Aegean Seas and various straits.

Since Bulgarian lev is anyway bounded to Euro Bulgaria is looking for more positive outcomes of the becoming a member of the Eurozone rather than risks.

In general, a linear mapping on a normed space is continuous if and only if it is bounded on the closed unit ball.

Note that the requirement that the maps be continuous is essential ; if X is infinite-dimensional, there exist linear maps which are not continuous, and therefore not bounded.

* The algebra of all bounded real-or complex-valued functions defined on some set ( with pointwise multiplication and the supremum norm ) is a unital Banach algebra.

* The algebra of all bounded continuous real-or complex-valued functions on some locally compact space ( again with pointwise operations and supremum norm ) is a Banach algebra.

* C *- algebra: A Banach algebra that is a closed *- subalgebra of the algebra of bounded operators on some Hilbert space.

When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

The Lie algebra of any compact Lie group ( very roughly: one for which the symmetries form a bounded set ) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones.

Here C < sub > b </ sub >( X ) denotes the C *- algebra of all continuous bounded functions on X with sup-norm.

The space of bounded linear operators B ( X ) on a Banach space X is an example of a unital Banach algebra.

This extends the definition for bounded linear operators B ( X ) on a Banach space X, since B ( X ) is a Banach algebra.

* Positive element of a C *- algebra ( such as a bounded linear operator ) whose spectrum consists of positive real numbers

In many situations in abstract algebra, such as for abelian groups, vector spaces or modules, the cokernel of the homomorphism f: X → Y is the quotient of Y by the image of f. In topological settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure of the image before passing to the quotient.

This suggests that one might define a noncommutative Riemannian manifold as a spectral triple ( A, H, D ), consisting of a representation of a C *- algebra A on a Hilbert space H, together with an unbounded operator D on H, with compact resolvent, such that is bounded for all a in some dense subalgebra of A.