Half ring (algebraic structure)
Half ring 
touches the specialties 
is a special case of 

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 noncommutative addition as well as with ( absorbing ) and / or , the definitions in the literature are not uniform.
Definitions
Half ring
A half ring (engl .: Semi ring ) is an algebraic structure with a (nonempty) amount and with two twodigit links ( addition ) and ( multiplication ), in which:
 is a commutative semigroup .
 is a half group.
 The distributive laws apply , i. H. for all true
 such as
If it is also commutative, one speaks of a commutative halfring.
Zero element
If a halfring 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 halfring.
The zero of a halfring 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 halfring 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 halfring.
A hemiring with a one is also called a valuation half ring .
Dioid
A hemiring with one and idempotent addition is called a dioid ; H. are at a dioid and u. a. Monoids .
Examples
 ;
 is even a half body .
 , the socalled MinPlusAlgebra ;
 For any set the power set is a half ring.
 More generally, every Boolean algebra is a half ring.
literature
 François Baccelli, Guy Cohen, Geert J. Olsder, JeanPierre Quadrat: Synchronization and Linearity (online version) . Wiley, New York 1992, ISBN 047193609X .
 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 0792357868 [4] .
 Udo Hebisch , Hanns J. Weinert: Half rings. Algebraic Theory and Applications in Computer Science . Teubner, Stuttgart 1993. ISBN 3519020912 .
Notes and individual references
 ↑ One also says: distributes over .
 ^ D. R. La Torre: On hideals and kideals in hemirings. Publ. Math. Debrecen 12, 219226 (1965) [1] [2] .
 ^ Hebisch, Weinert; P. 257
Web links
 Fun with Semirings (PDF; 252 kB)