Half ring (algebraic structure)

Half ring

touches the specialties

mathematics Abstract algebra Group theory Number theory

is a special case of

Left half ring

includes as special cases

Boolean algebra Dioid (m. E., see left) Half body ${\displaystyle (}$natural numbers ${\displaystyle ,+,\cdot )}$ ring

In mathematics, a half ring is the generalization of the algebraic structure of a ring , in which the addition no longer has to be a commutative group , but only a commutative semigroup .

Half rings are also defined with non-commutative addition as well as with ( absorbing ) and / or , the definitions in the literature are not uniform. ${\displaystyle 0}$ ${\displaystyle 1}$

Definitions

Half ring

A half ring (engl .: Semi ring ) is an algebraic structure with a (non-empty) amount and with two two-digit links ( addition ) and ( multiplication ), in which: ${\displaystyle (H,+,\cdot )}$ ${\displaystyle H}$ ${\displaystyle +\colon H\times H\to H}$ ${\displaystyle \cdot \colon H\times H\to H}$

1. ${\displaystyle (H,+)}$is a commutative semigroup .
2. ${\displaystyle (H,\cdot )}$ is a half group.
3. The distributive laws apply , i. H. for all true ${\displaystyle a,b,c\in H}$

${\displaystyle (a+b)\cdot c=a\cdot c+b\cdot c}$   such as   ${\displaystyle c\cdot (a+b)=c\cdot a+c\cdot b.}$

If it is also commutative, one speaks of a commutative half-ring. ${\displaystyle (H,\cdot )}$

Zero element

If a half-ring has a neutral element with respect to addition, i. H. ${\displaystyle (H,+,\cdot )}$ ${\displaystyle 0\in H}$

${\displaystyle 0+a=a+0=a}$ for all ${\displaystyle a\in H,}$

so this is called the zero element or, for short, the zero of the half-ring.

The zero of a half-ring is said to be absorbing , if related to the multiplication, i.e. H. ${\displaystyle 0}$

${\displaystyle 0\cdot a=a\cdot 0=0}$ for all ${\displaystyle a\in H.}$

A half ring with an absorbing zero is also called a hemiring . ${\displaystyle (H,+,0,\cdot )}$

One element

If a half-ring contains a neutral element with respect to multiplication, so ${\displaystyle 1\in H}$

${\displaystyle 1\cdot a=a\cdot 1=a}$ for all ${\displaystyle a\in H,}$

then this is called the one element or, for short, the one of the half-ring.

A hemiring with a one is also called a valuation half- ring . ${\displaystyle (H,+,0,\cdot ,1)}$ ${\displaystyle 1\neq 0}$

Dioid

A hemiring with one and idempotent addition is called a dioid ; H. are at a dioid and u. a. Monoids . ${\displaystyle (D,+,0,\cdot ,1)}$ ${\displaystyle (D,+,0)}$ ${\displaystyle (D,\cdot ,1)}$

Examples

• ${\displaystyle (\mathbb {N} ,+,0,\cdot ,1)}$;
• ${\displaystyle (\mathbb {Q} _{+},+,0,\cdot ,1)}$is even a half body .
• ${\displaystyle (\mathbb {R} \cup \{\infty \},\operatorname {min} ,\infty ,+,0)}$, the so-called Min-Plus-Algebra ;
• For any set the power set is a half ring. ${\displaystyle X}$ ${\displaystyle ({\mathcal {P}}(X),\cup ,\emptyset ,\cap ,X)}$
• More generally, every Boolean algebra is a half ring.

literature

• François Baccelli, Guy Cohen, Geert J. Olsder, Jean-Pierre Quadrat: Synchronization and Linearity (online version) . Wiley, New York 1992, ISBN 0-471-93609-X .
• Jonathan S. Golan: 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 [3] . Kluwer Academic Publishers, Dordrecht 1999. ISBN 0-7923-5786-8 [4] .
• Udo Hebisch , Hanns J. Weinert: Half rings. Algebraic Theory and Applications in Computer Science . Teubner, Stuttgart 1993. ISBN 3-519-02091-2 .

Notes and individual references

1. ↑ One also says: distributes over . ${\displaystyle \cdot }$ ${\displaystyle +}$
2. ^ D. R. La Torre: On h-ideals and k-ideals in hemirings. Publ. Math. Debrecen 12, 219-226 (1965) [1] [2] .
3. ^ Hebisch, Weinert; P. 257

Web links

• Fun with Semirings (PDF; 252 kB)