Difference between revisions of "Semi-ring"
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| + | A non-empty set $S$ with two associative binary operations $+$ and $\cdot$, satisfying the [[distributive law]]s | ||
| + | $$ | ||
| + | (a+b) \cdot c = a\cdot c + b \cdot c | ||
| + | $$ | ||
and | and | ||
| + | $$ | ||
| + | a \cdot (b+c) = a\cdot b + a\cdot c \ . | ||
| + | $$ | ||
| + | In most cases one also assumes that the addition is commutative and that there exists a zero element $0$ such that $a + 0 = a$ for every $a \in S$. The most important classes of semi-rings are [[ring]]s and [[distributive lattice]]s. If there is a multiplicative [[unit element]] 1, the two classes are combined by the condition | ||
| + | $$ | ||
| + | \forall x \, \exists y \ x+y=1 \ . | ||
| + | $$ | ||
| + | The non-negative integers with the usual operations provide an example of a semi-ring that does not satisfy this condition. | ||
| − | + | ====Comments==== | |
| + | The term "exotic" semi-rings has been used to describe subsets of the real numbers with $\min$ or $\max$ as ${+}$ and addition as ${\star}$. These are thus [[idempotent semi-ring]]s. Examples include the ''tropical'' semiring on $\mathbf{N} \cup \{\infty\}$ with operations ${\min},\, +$. | ||
| − | + | An ''additive zero'' in a semiring $S$ is an element $a$ such that $a+x = x+a = x$ for all $x$; a ''multiplicative zero'' is an element $m$ such that $m \cdot x = x \cdot m = m$ for all $x$. A ''double zero'' is an element which is both an additive zero and a multiplicative zero. | |
| − | + | If the additive semigroup of a semiring $S$ is commutative and satisfies the [[Semi-group with cancellation|cancellative property]] $a + c = b + c \Rightarrow a = b$ for all $c$, then the additive semigroup embeds in its [[Grothendieck group]] $R$ and the multiplication $\cdot$ extends to $R$, giving it a ring structure: the ''Grothendieck ring'' of $S$. The Grothendieck ring of a finite group $G$ over a field $K$ is the ring constructed in this way from the semiring of isomorphism classes of modules over the [[group ring]] $K[G]$ with [[direct sum]] and [[tensor product]] as the operations. | |
| − | + | ====References==== | |
| + | * K. Glazek, ''A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences: With Complete Bibliography'' Springer (2013) {{ISBN|9401599645}} | ||
| + | * U. Hebisch, H.J. Weinert, ''Semirings: Algebraic Theory and Applications in Computer Science'' World Scientific (1998) {{ISBN|9814495697}} {{ZBL|0934.16046}} | ||
| + | * Serge Lang ''Algebra'' (3rd rev. ed.) Graduate Texts in Mathematics '''211''' Springer (2002) {{ZBL|0984.00001}} | ||
Latest revision as of 05:54, 15 April 2023
2020 Mathematics Subject Classification: Primary: 16Y60 [MSN][ZBL]
A non-empty set $S$ with two associative binary operations $+$ and $\cdot$, satisfying the distributive laws $$ (a+b) \cdot c = a\cdot c + b \cdot c $$ and $$ a \cdot (b+c) = a\cdot b + a\cdot c \ . $$ In most cases one also assumes that the addition is commutative and that there exists a zero element $0$ such that $a + 0 = a$ for every $a \in S$. The most important classes of semi-rings are rings and distributive lattices. If there is a multiplicative unit element 1, the two classes are combined by the condition $$ \forall x \, \exists y \ x+y=1 \ . $$ The non-negative integers with the usual operations provide an example of a semi-ring that does not satisfy this condition.
Comments
The term "exotic" semi-rings has been used to describe subsets of the real numbers with $\min$ or $\max$ as ${+}$ and addition as ${\star}$. These are thus idempotent semi-rings. Examples include the tropical semiring on $\mathbf{N} \cup \{\infty\}$ with operations ${\min},\, +$.
An additive zero in a semiring $S$ is an element $a$ such that $a+x = x+a = x$ for all $x$; a multiplicative zero is an element $m$ such that $m \cdot x = x \cdot m = m$ for all $x$. A double zero is an element which is both an additive zero and a multiplicative zero.
If the additive semigroup of a semiring $S$ is commutative and satisfies the cancellative property $a + c = b + c \Rightarrow a = b$ for all $c$, then the additive semigroup embeds in its Grothendieck group $R$ and the multiplication $\cdot$ extends to $R$, giving it a ring structure: the Grothendieck ring of $S$. The Grothendieck ring of a finite group $G$ over a field $K$ is the ring constructed in this way from the semiring of isomorphism classes of modules over the group ring $K[G]$ with direct sum and tensor product as the operations.
References
- K. Glazek, A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences: With Complete Bibliography Springer (2013) ISBN 9401599645
- U. Hebisch, H.J. Weinert, Semirings: Algebraic Theory and Applications in Computer Science World Scientific (1998) ISBN 9814495697 Zbl 0934.16046
- Serge Lang Algebra (3rd rev. ed.) Graduate Texts in Mathematics 211 Springer (2002) Zbl 0984.00001
Semi-ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-ring&oldid=16685