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In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed. There are at least two conflicting conventions of what constitutes a semifield. * In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is the analogue of a division algebra, but defined over the integers Z rather than over a field.〔 More precisely, it is a Z-algebra whose nonzero elements form a loop under multiplication. In other words, a semifield is a set ''S'' with two operations + (addition) and · (multiplication), such that * * (''S'',+) is an abelian group, * * multiplication is distributive on both the left and right, * * there exists a multiplicative identity element, and * * division is always possible: for every ''a'' and every nonzero ''b'' in ''S'', there exist unique ''x'' and ''y'' in ''S'' for which ''b''·''x'' = ''a'' and ''y''·''b'' = ''a''. : Note in particular that the multiplication is not assumed to be commutative or associative. A semifield that is associative is a division ring, and one that is both associative and commutative is a field. A semifield by this definition is a special case of a quasifield. If ''S'' is finite, the last axiom in the definition above can be replaced with the assumption that there are no zero divisors, so that ''a''·''b'' = 0 implies that ''a'' = 0 or ''b'' = 0.〔 Note that due to the lack of associativity, the last axiom is ''not'' equivalent to the assumption that every nonzero element has a multiplicative inverse, as is usually found in definitions of fields and division rings. * In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (''S'',+,·) in which all elements have a multiplicative inverse.〔〔 These objects are also called proper semifields. A variation of this definition arises if ''S'' contains an absorbing zero that is different from the multiplicative unit ''e'', it is required that the non-zero elements be invertible, and ''a''·0 = 0·''a'' = 0. Since multiplication is associative, the (non-zero) elements of a semifield form a group. However, the pair (''S'',+) is only a semigroup, i.e. additive inverse need not exist, or, colloquially, 'there is no subtraction'. Sometimes, it is not assumed that the multiplication is associative. == Primitivity of Semifields== A semifield D is called right (resp. left) primitive if it has an element w such that the set of nonzero elements of D * is equal to the set of all right (resp. left) principal powers of w. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Semifield」の詳細全文を読む スポンサード リンク
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