* 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 )

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

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

# 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 ).

The category of spectral spaces which has spectral maps as morphisms is dually equivalent to the category of bounded distributive lattices ( together with morphisms of such lattices.

However, in a bounded distributive lattice every element will have at most one complement.

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

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

* 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.

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.

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.

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

Having volunteered that he was a man of about sixty, he bounded up the stairs and with each leap rendered the number less credible.

Here K denotes the field of real numbers or complex numbers, I is a closed and bounded interval b and p, q are real numbers with 1 < p, q < ∞ so that

In computational complexity theory, BQP ( bounded error quantum polynomial time ) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1 / 3 for all instances.

* 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.

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.

The larger, northern portion of the basin is bounded within Chad by the Tibesti Mountains in the northwest, the Ennedi Plateau in the northeast, the Ouaddaï Highlands in the east along the border with Sudan, the Guéra Massif in central Chad, and the Mandara Mountains along Chad's southwestern border with Cameroon.

For instance, any continuous function defined on a compact space into an ordered set ( with the order topology ) such as the real line is bounded.

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

Computationally, a context-sensitive language is equivalent with a linear bounded nondeterministic Turing machine, also called a linear bounded automaton.

In turn from PIE * keu ( b )-, " to bend, turn ".</ ref > is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In the case of C *- algebras, any *- homomorphism π between C *- algebras is non-expansive, i. e. bounded with norm ≤ 1.

The space C of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm.

Largely complete by 1230, the castle was of typical Norman courtyard design, with a central square without a keep, bounded on all sides by tall defensive walls and protected at each corner by a circular tower.

* A " stable " filter produces an output that converges to a constant value with time, or remains bounded within a finite interval.

An " unstable " filter can produce an output that grows without bounds, with bounded or even zero input.

Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection.

At each end is a goal 2. 14 m ( 7 feet ) high and 3. 66 m ( 12 ft ) wide measured from the inner sides of the posts and crossbar, and an approximately semi-circular area 14. 63 m ( 16 yd ) from the goal known as the shooting circle ( or D or arc ), bounded by a solid line, with a dotted line 5 m ( 5 yd 6 in — this marking was not established until after metric conversion ) from that, as well as lines across the field 22. 90 m ( 25 yd ) from each end-line ( generally referred to as the 23 m lines ) and in the center of the field.

When applied on an image to fill a particular bounded area with color, it is also known as boundary fill.

A basic distinction is with regard to whether the speaker looks at a situation as bounded and unitary, without reference to any flow of time during the situation (" I ate "), or with no reference to temporal bounds but with reference to the nature of the flow of time during the situation (" I was eating ", " I used to eat ").

Differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change while integral calculus is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines and, such that areas above the axis add to the total, and the area below the x axis subtract from the total.

* π < sub > T </ sub > is an involution preserving and unit-preserving homomorphism from the ring of complex-valued bounded measurable functions on R.