Half ring (algebraic structure)

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Half ring

touches the specialties

is a special case of

  • Left half ring

includes as special cases

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.


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:

  1. is a commutative semigroup .
  2. is a half group.
  3. The distributive laws apply , i. H. for all true
  such as  

If it is also commutative, one speaks of a commutative half-ring.

Zero element

If a half-ring has a neutral element with respect to addition, i. H.

for all

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.

for all

A half ring with an absorbing zero is also called a hemiring .

One element

If a half-ring contains a neutral element with respect to multiplication, so

for all

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 .


A hemiring with one and idempotent addition is called a dioid ; H. are at a dioid and u. a. Monoids .


  • ;
  • is even a half body .
  • , the so-called Min-Plus-Algebra ;
  • For any set the power set is a half ring.
  • More generally, every Boolean algebra is a half ring.


  • 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 .
  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

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