Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.

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

* Every Boolean algebra is a distributive lattice.

* Every Heyting algebra is a distributive lattice.

* Every totally ordered set is a distributive lattice with max as join and min as meet.

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

Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients.

* Every substructure is the union of its finitely generated substructures ; hence Sub ( A ) is an algebraic lattice.

Also, a kind of converse holds: Every algebraic lattice is isomorphic to Sub ( A ) for some algebra A.

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

Every distributive lattice is modular.

Every non-modular lattice contains a copy of N < sub > 5 </ sub > as a sublattice.

Every quasinormal subgroup is a modular subgroup, that is, a modular element in the lattice of subgroups.

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

: Every non-empty set A contains an element B which is disjoint from A.

Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.

Every element of is a member of the equivalence class.

Every repetition of insertion sort removes an element from the input data, inserting it into the correct position in the already-sorted list, until no input elements remain.

Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms.

Every simple module is cyclic, that is it is generated by one element.

Every element s, except a possible greatest element, has a unique successor ( next element ), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound.

Every element has a successor ( there is no largest element ).

* Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element with respect to set inclusion.

Every time a pixel on a triangle is rendered, the corresponding texel ( or texture element ) in the texture must be found.

Every non-inner automorphism yields a non-trivial element of Out ( G ), but different non-inner automorphisms may yield the same element of Out ( G ).

Every finite tree structure has a member that has no superior.

Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes ( or congruence classes ) for the relation.

Every group has a presentation, and in fact many different presentations ; a presentation is often the most compact way of describing the structure of the group.

Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation ( s ) defining the structure.

Every solid has its own characteristic energy band structure.

Every chapter contains a comprehensive compilation all of the published examples of the reaction organized in tables according to the structure of the starting material.

Every Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions.

: Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.

* " What Every Programmer Should Know About Memory " by Ulrich Drepper — explains the structure of modern memory subsystems and suggests how to utilize them efficiently

Every H * is very special in structure: it is pure-injective ( also called algebraically compact ), which says more or less that solving equations in H * is relatively straightforward.

Every structure is associated with a certain quantity of energy, which determines the stability of the molecule or ion ( the lower energy, the greater stability ).

Every Hermitian manifold is a complex manifold which comes naturally equipped with a Hermitian form and an integrable, almost complex structure.

Every animal cell is enclosed in a plasma membrane, which has the structure of a lipid bilayer with many types of large molecules embedded in it.

Every structure from the World's Columbian Exposition was long ago destroyed by fire, demolished or moved elsewhere, except for the old Palace of Fine Arts, now the Museum of Science and Industry, The Palace of Fine Arts, the only fireproof building at the fair, fell into disrepair and was rehabilitated with a $ 5 million grant in 1930 from Julius Rosenwald ( President of Sears, Roebuck and Co .).

Every five years, starting the day after Labor Day, the New Jersey Department of Environmental Protection ’ s ( DEP ) water level management plan allows the lake to be lowered five feet to allow for inspection of the dam structure, property owners ’ repairs to lakeshore structures, aquatic vegetation control, and silt and drainage material removal from areas around the lake.

Every timber structure on the site had been burnt, the charcoal being the only organic matter that survived the acid soils.

Every server uses its own query logic and structure.

Every ( normal ) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it is isomorphic to the complex algebra corresponding to the field.

Every compact smooth manifold of dimension 2n, which has only handles of index ≤ n, has a Stein structure provided n > 2, and when n = 2 the same holds provided the 2-handles are attached with certain framings ( framing less than the Thurston-Bennequin framing ).

Every window was lit, to lurid effect, as if the interior of the structure might be on fire.

Every Menards store shares a common structure.

Every structure shown here was built during this period, along with tens of others