# Half ring (algebraic structure)

 Half ring touches the specialties mathematics 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