A Heyting field is one of the inequivalent ways in constructive mathematics to capture the classical notion of a field. It is essentially a field with...

Arend Heyting (Dutch: [ˈaːrənt ˈɦɛitɪŋ]; 9 May 1898 – 9 July 1980) was a Dutch mathematician and logician. Heyting was a student of Luitzen Egbertus Jan...

algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced in 1930 by Arend Heyting to formalize...

(1984), Chapter 3 Mines, Richman & Ruitenburg (1988), §II.2. See also Heyting field. Beachy & Blair (2006), p. 120, Ch. 3 Artin (1991), Chapter 13.4 Lidl...

the topology. Every Heyting algebra can be represented by a topological field of sets with the underlying lattice of the Heyting algebra corresponding...

field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is...

analysis. Constructive frameworks for its formulation are extensions of Heyting arithmetic by types including N N {\displaystyle {\mathbb {N} }^{\mathbb...

ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are...

and complex numbers. Scalars can also be, more generally, elements of any field. Vector spaces generalize Euclidean vectors, which allow modeling of physical...

Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra. It is important to compare the...

in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see...

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic...

open elements of an interior algebra form a Heyting algebra and the closed elements form a dual Heyting algebra. The regular open elements and regular...

In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined...

Hilbert. Brouwer's ideas were subsequently taken up by his student Arend Heyting and Hilbert's former student Hermann Weyl. In addition to his mathematical...

involves a second structure called a field, and an operation called scalar multiplication between elements of the field (called scalars), and elements of...

⊃ principal ideal domains ⊃ euclidean domains ⊃ fields ⊃ algebraically closed fields Formally, a unique factorization domain is defined to be...

⊃ principal ideal domains ⊃ euclidean domains ⊃ fields ⊃ algebraically closed fields A ring is a set R equipped with two binary operations + (addition)...

together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal...

Boolean algebra Monadic Boolean algebra De Morgan algebra First-order logic Heyting algebra Lindenbaum–Tarski algebra Skew Boolean algebra Algebraic normal...

If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra. A chain...

⊃ principal ideal domains ⊃ euclidean domains ⊃ fields ⊃ algebraically closed fields An integral domain is a nonzero commutative ring in which...

the definition: see below. A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however...

factorization domain. Any field, including the fields of rational numbers, real numbers, and complex numbers, is Noetherian. (A field only has two ideals —...

Examples: A graded vector space is an example of a graded module over a field (with the field having trivial grading). A graded ring is a graded module over itself...

Knaster–Tarski theorem Infinite divisibility Heyting algebra Relatively complemented lattice Complete Heyting algebra Pointless topology MV-algebra Ockham...

specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales).[citation needed] A complete lattice is a partially...

a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept...

principal ideal domains ⊃ euclidean domains ⊃ fields ⊃ algebraically closed fields Examples include: K {\displaystyle K} : any field, Z {\displaystyle \mathbb {Z} }...

a complete lattice. Complete Heyting algebra. A Heyting algebra that is a complete lattice is called a complete Heyting algebra. This notion coincides...