Semiring

Prejudice to the special areas

  • 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
  • Natural numbers
  • Ring

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

Semirings are defined as non commutative addition and with ( absorptive ) and / or the definitions in the literature are not uniform.

Definitions

Semiring

A semiring (English: semiring ) is an algebraic structure with a (non- empty) set and with two double-digit shortcuts (addition) and (multiplication), for which:

Is also commutative, then one speaks of a commutative semiring.

Zero element

If a half-ring is, a neutral element with respect to addition, d

Then we call this the zero element or the zero of the short half-ring. The zero half ring is called absorbing if

A semiring with an absorbing zero also means Hemiring.

One element

When a half-ring contains a neutral element with respect to multiplication, ie

Then this is called the identity element or just the one half of the ring.

A Hemiring with a one half-ring is also called evaluation.

Dioid

A Hemiring with one and idempotent addition is called Dioid, ie are in a Dioid and inter alia monoids.

Examples

  • ;
  • Is even a half-body.
  • , Called the Min -Plus Algebra;
  • For each set, the power set is a semi- ring.
  • General every Boolean algebra is a semiring.
Absorbing element
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