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Viewing Version
4
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'semiring'
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| Title of object: |
semiring |
| Canonical Name: |
Semiring |
| Type: |
Definition |
| Created on: |
2002-02-24 18:05:24 |
| Modified on: |
2004-02-21 04:20:51 |
| Classification: |
msc:16Y60 |
| Keywords: |
partial order, poset |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts} |
Content:
\PMlinkescapeword{algebra}
\PMlinkescapeword{constants}
A \emph{semiring} is an algebra $(A, \cdot, +, 0, 1)$ of a set $A$, where 0 and 1 are constants, $(A, \cdot, 1)$ is a monoid, $(A, +, 0)$ is a commutative monoid, $\cdot$ \PMlinkname{distributes}{Distributivity} over $+$ from the left and right, and 0 is both a left and right annihilator ($0a = a0 = 0$).
Often $a\cdot b$ is written simply as $ab$, and the semiring $(A, \cdot, +, 0, 1)$ as simply $A$.
The \PMlinkname{relation}{Relation} $\leq$ on a semiring $A$ is defined as $a \leq b$ if and only if there exists some $c\in A$ such that $a + c = b$, and is a quasiordering.
If $+$ is \PMlinkname{idempotent}{Idempotency} over $A$ (that is, $a + a = a$ holds for all $a\in A$), then $\leq$ is a partial order.
Addition and (left and right) multiplication are monotonic operators with respect to $\leq$, with 0 as the minimal element. |
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