Semifield: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 05:01, 18 June 2024

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.

Overview

The term semifield has two conflicting meanings, both of which include fields as a special case.

In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is a nonassociative division ring with multiplicative identity element.^[1] More precisely, it is a nonassociative ring 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.^[2] 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 nonzero elements have a multiplicative inverse.^[3]^[4] 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.

Examples

We only give examples of semifields in the second sense, i.e. additive semigroups with distributive multiplication. Moreover, addition is commutative and multiplication is associative in our examples.

Positive rational numbers with the usual addition and multiplication form a commutative semifield.
This can be extended by an absorbing 0.
Positive real numbers with the usual addition and multiplication form a commutative semifield.
This can be extended by an absorbing 0, forming the probability semiring, which is isomorphic to the log semiring.
Rational functions of the form f /g, where f and g are polynomials over a subfield of real numbers in one variable with positive coefficients, form a commutative semifield.
This can be extended to include 0.
The real numbers R can be viewed a semifield where the sum of two elements is defined to be their maximum and the product to be their ordinary sum; this semifield is more compactly denoted (R, max, +). Similarly (R, min, +) is a semifield. These are called the tropical semiring.
This can be extended by −∞ (an absorbing 0); this is the limit (tropicalization) of the log semiring as the base goes to infinity.
Generalizing the previous example, if (A,·,≤) is a lattice-ordered group then (A,+,·) is an additively idempotent semifield with the semifield sum defined to be the supremum of two elements. Conversely, any additively idempotent semifield (A,+,·) defines a lattice-ordered group (A,·,≤), where a≤b if and only if a + b = b.
The boolean semifield B = {0, 1} with addition defined by logical or, and multiplication defined by logical and.

References

^ Donald Knuth, Finite semifields and projective planes. J. Algebra, 2, 1965, 182--217 MR 0175942.
^ Landquist, E.J., "On Nonassociative Division Rings and Projective Planes", Copyright 2000.
^ Golan, Jonathan S., Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Sci. Tech., Harlow, 1992, MR1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 MR1746739.
^ Hebisch, Udo; Weinert, Hanns Joachim, Semirings and semifields. Handbook of algebra, Vol. 1, 425--462, North-Holland, Amsterdam, 1996. MR1421808.

[Knuth-1] Donald Knuth, Finite semifields and projective planes. J. Algebra, 2, 1965, 182--217 MR 0175942.

[Landquist-2] Landquist, E.J., "On Nonassociative Division Rings and Projective Planes", Copyright 2000.

[Golan-3] Golan, Jonathan S., Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Sci. Tech., Harlow, 1992, MR1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 MR1746739.

[HW-4] Hebisch, Udo; Weinert, Hanns Joachim, Semirings and semifields. Handbook of algebra, Vol. 1, 425--462, North-Holland, Amsterdam, 1996. MR1421808.

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
+{{Short description|Algebraic structure}}
-In [[mathematics]],  a '''semifield''' is an [[algebraic structure]] with two [[binary operation]]s, addition and multiplication, which is similar to a [[field (mathematics)|field]], but with some axioms relaxed. There are at least two conflicting conventions of what constitutes a semifield.
+{{Ring theory sidebar}}
+In [[mathematics]], a '''semifield''' is an [[algebraic structure]] with two [[binary operation]]s, addition and multiplication, which is similar to a [[field (mathematics)|field]], but with some axioms relaxed.
+==Overview==
-* In [[projective geometry]] and [[finite geometry]] ([[Mathematics Subject Classification|MSC]] 51A, 51E, 12K10), a '''semifield''' is the analogue of a [[division algebra]], but defined over the [[integer]]s '''Z''' rather than over a field.<ref name="Knuth" /> More precisely, it is a '''Z'''-algebra whose nonzero elements form a [[loop (mathematics)|loop]] under multiplication. In other words, a semifield is a set ''S'' with two operations + (addition) and · (multiplication), such that
+The term semifield has two conflicting meanings, both of which include fields as a special case.
-** (''S'',+) is an [[abelian group]],
+* In [[projective geometry]] and [[finite geometry]] ([[Mathematics Subject Classification|MSC]] 51A, 51E, 12K10), a '''semifield''' is a [[Division_algebra#Not_necessarily_associative_division_algebras|nonassociative division ring]] with multiplicative identity element.<ref name="Knuth" />  More precisely, it is a [[nonassociative algebra|nonassociative ring]] whose nonzero elements form a [[loop (algebra)|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 property|distributive]] on both the left and right,
 ** there exists a multiplicative [[identity element]], and
 ** [[division (mathematics)|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 property|commutative]] or [[associative property|associative]]. A semifield that is associative is a [[division ring]], and one that is both associative and commutative is a [[field (mathematics)|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 divisor]]s, so that ''a''&middot;''b'' = 0 implies that ''a'' = 0 or ''b'' = 0.<ref name="Landquist" /> 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.
+: Note in particular that the multiplication is not assumed to be [[commutative property|commutative]] or [[associative property|associative]]. A semifield that is associative is a [[division ring]], and one that is both associative and commutative is a [[field (mathematics)|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 divisor]]s, so that ''a''⋅''b'' = 0 implies that ''a'' = 0 or ''b'' = 0.<ref name="Landquist" /> 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]] ([[Mathematics Subject Classification|MSC]] 16Y60), a '''semifield''' is a [[semiring]] (''S'',+,·) in which all elements have a multiplicative inverse.<ref name="Golan" /><ref name="HW" /> 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 (mathematics)|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.
+* In [[ring theory]], [[combinatorics]], [[functional analysis]], and [[theoretical computer science]] ([[Mathematics Subject Classification|MSC]] 16Y60), a '''semifield''' is a [[semiring]] (''S'',+,·) in which all nonzero elements have a multiplicative inverse.<ref name="Golan" /><ref name="HW" /> 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 (mathematics)|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==
+== 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.
@@ Line 16: / Line 21: @@
 We only give examples of semifields in the second sense, i.e. additive semigroups with distributive multiplication. Moreover, addition is commutative and multiplication is associative in our examples.
+* [[Sign (mathematics)|Positive]] [[rational numbers]] with the usual addition and multiplication form a commutative semifield.
+*:This can be extended by an absorbing 0.
 * [[Positive real numbers]] with the usual addition and multiplication form a commutative semifield.
+*:This can be extended by an absorbing 0, forming the [[probability semiring]], which is isomorphic to the [[log semiring]].
-* [[Rational function]]s of the form ''f'' /''g'', where ''f'' and ''g'' are [[polynomial]]s in one variable with positive coefficients form a commutative semifield.
+* [[Rational function]]s of the form ''f'' /''g'', where ''f'' and ''g'' are [[polynomial]]s over a subfield of real numbers in one variable with positive coefficients, form a commutative semifield.
-* [[Max-plus algebra]], or the [[tropical semiring]], ('''R''', max, +) is a semifield. Here the sum of two elements is defined to be their maximum, and the product to be their ordinary sum.
+*:This can be extended to include 0.
-* If (''A'',≤) is a [[lattice ordered group]] then (''A'',+,·) is an additively idempotent semifield. The semifield sum is defined to be the sup of two elements. Conversely, any additively idempotent semifield (''A'',+,·) defines a lattice-ordered group (''A'',≤), where ''a''≤''b'' if and only if ''a'' + ''b'' = ''b''.
+* The [[real numbers]] '''R''' can be viewed a semifield where the sum of two elements is defined to be their maximum and the product to be their ordinary sum; this semifield is more compactly denoted ('''R''', max, +).  Similarly ('''R''', min, +) is a semifield. These are called the [[tropical semiring]].
+*:This can be extended by −∞ (an absorbing 0); this is the limit ([[tropicalization]]) of the [[log semiring]] as the base goes to infinity.
+* Generalizing the previous example, if (''A'',·,≤) is a [[Lattice ordered group|lattice-ordered group]] then (''A'',+,·) is an additively [[idempotent]] semifield with the semifield sum defined to be the [[supremum]] of two elements. Conversely, any additively idempotent semifield (''A'',+,·) defines a lattice-ordered group (''A'',·,≤), where ''a''≤''b'' if and only if ''a'' + ''b'' = ''b''.
+* The boolean semifield '''B''' = {0, 1} with addition defined by [[logical or]], and multiplication defined by [[logical and]].
 ==See also==
@@ Line 30: / Line 41: @@
 <ref name="Landquist">Landquist, E.J., "On Nonassociative Division Rings and Projective Planes", Copyright 2000.</ref>
 <ref name="Knuth">[[Donald Knuth]], ''Finite semifields and projective planes''. J. Algebra, 2, 1965, 182--217 {{MathSciNet|id=0175942}}.</ref>
-<ref name="Golan">Golan, Jonathan S., ''Semirings and their applications''. Updated and expanded version of ''The theory of semirings, with applications to mathematics and theoretical computer science'' (Longman Sci. Tech., Harlow, 1992, {{MathSciNet|id=1163371}}. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 {{MathSciNet|id=1746739}}.</ref>
+<ref name="Golan">Golan, Jonathan S., ''Semirings and their applications''. Updated and expanded version of ''The theory of semirings, with applications to mathematics and theoretical computer science'' (Longman Sci. Tech., Harlow, 1992, {{MathSciNet|id=1163371}}. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. {{isbn|0-7923-5786-8}} {{MathSciNet|id=1746739}}.</ref>
 <ref name="HW">Hebisch, Udo; Weinert, Hanns Joachim, ''Semirings and semifields''. Handbook of algebra, Vol. 1, 425--462, North-Holland, Amsterdam, 1996. {{MathSciNet|id=1421808}}.</ref>
 }}
 [[Category:Algebraic structures]]
+[[Category:Ring theory]]