In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum ( join ) and an infimum ( meet ).

In mathematics, Minkowski's theorem is the statement that any convex set in R < sup > n </ sup > which is symmetric with respect to the origin and with volume greater than 2 < sup > n </ sup > d ( L ) contains a non-zero lattice point.

* Bottom element, in lattice theory and related branches of mathematics

Napier's bones is an abacus created by John Napier for calculation of products and quotients of numbers that was based on Arab mathematics and lattice multiplication used by Matrakci Nasuh in the Umdet-ul Hisab and Fibonacci writing in the Liber Abaci.

In mathematics, E < sub > 6 </ sub > is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras, all of which have dimension 78 ; the same notation E < sub > 6 </ sub > is used for the corresponding root lattice, which has rank 6.

** Join ( mathematics ), a least upper bound of set orders in lattice theory

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 his article Restructuring Lattice Theory ( 1982 ) initiating formal concept analysis as a mathematical discipline, Rudolf Wille starts from a discontent with the current lattice theory and pure mathematics in general: The production of theoretical results-often achieved by " elaborate mental gymnastics "-were impressive, but the connections between neighbouring domains, even parts of a theory were getting weaker.

In mathematics, the Leech lattice is an even unimodular lattice Λ < sub > 24 </ sub > in 24-dimensional Euclidean space E < sup > 24 </ sup > found by.

This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory.

In mathematics, especially in geometry and group theory, a lattice in is a discrete subgroup of which spans the real vector space.

They also arise in applied mathematics in connection with coding theory, in cryptography because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences.

The period lattice in is central to the study of elliptic functions, developed in nineteenth century mathematics ; it generalises to higher dimensions in the theory of abelian functions.

In mathematics, a lattice is a partially ordered set in which any two elements have a supremum ( also called a least upper bound or join ) and an infimum ( also called a greatest lower bound or meet ).

In mathematics, E < sub > 8 </ sub > is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248 ; the same notation is used for the corresponding root lattice, which has rank 8.

In mathematics, E < sub > 7 </ sub > is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e < sub > 7 </ sub >, all of which have dimension 133 ; the same notation E < sub > 7 </ sub > is used for the corresponding root lattice, which has rank 7.

In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice.

* E < sub > 8 </ sub >, a Lie group in mathematics, an exceptional simple Lie group with root lattice of rank 8

In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Julius Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice.

In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:

* Niemeier lattice, in mathematics

In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and corresponding bras and kets are related by the Riesz representation theorem.

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem.

In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem.

* Crystallographic restriction theorem, in mathematics

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below.

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups ( and their action on other mathematical objects ) can sometimes be reduced to questions about finite simple groups.

Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development.

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

Gödel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics.

The ineffectiveness of the completeness theorem can be measured along the lines of reverse mathematics.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

In mathematics, the Hahn – Banach theorem is a central tool in functional analysis.

Of course, our understanding of what the theorem really means gains in profundity as the mathematics around the theorem grows.

In mathematics, the Poincaré conjecture ( ; ) is a theorem about the characterization of the three-dimensional sphere ( 3-sphere ), which is the hypersphere that bounds the unit ball in four-dimensional space.

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.

On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics.

Some, on the other hand, may be called " deep ": their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics.

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order ( number of elements ) of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange.