Unital (other)
Unital may refer to: * A unital algebra – an algebra that contains a multiplicative identity element. * A geometric unital – a block design for integer . * A unital algebraic structure, such as a unital magma. * A unital map In abstract algebra, a unital map on a C*-algebra is a map \phi which preserves the identity element: :\phi ( I ) = I. This condition appears often in the context of completely positive maps, especially when they represent quantum operations. ...

Unital Algebra
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 structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity inst ...
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Unital (geometry)
In geometry, a unital is a set of ''n''3 + 1 points arranged into subsets of size ''n'' + 1 so that every pair of distinct points of the set are contained in exactly one subset. This is equivalent to saying that a unital is a 2-(''n''3 + 1, ''n'' + 1, 1) block design. Some unitals may be embedded in a projective plane of order ''n''2 (the subsets of the design become sets of collinear points in the projective plane). In this case of ''embedded unitals'', every line of the plane intersects the unital in either 1 or ''n'' + 1 points. In the Desarguesian planes, PG(2,''q''2), the classical examples of unitals are given by nondegenerate Hermitian curves. There are also many non-classical examples. The first and the only known unital with non prime power parameters, ''n''=''6'', was constructed by Bhaskar Bagchi and Sunanda Bagchi. It is still unknown if this unital can be embedded in a projective plane of order ''36'', if such a plane exists. Unitals Classical We review some termi ...
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Algebraic Structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called '' scalars''), and elements of the vector space (called '' vectors''). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type (homomor ...
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