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Compared to a ring , a semiring omits the requirement for inverses under addition; that is, it requires only a commutative monoid , not a commutative group. In a ring, the additive inverse requirement implies the existence of a multiplicative zero, so here it must be specified explicitly. This makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly.

These authors often use rig for the concept defined here. In particular, one can generalise the theory of associative algebras over commutative rings directly to a theory of algebras over commutative semirings. Then a ring is simply an algebra over the commutative semiring Z of integers.

These dynamic programming algorithms rely on the distributive property of their associated semirings to compute quantities over a large possibly exponential number of terms more efficiently than enumerating each of them.

A motivating example of a semiring is the set of natural numbers N including zero under ordinary addition and multiplication. Likewise, the non-negative rational numbers and the non-negative real numbers form semirings. All these semirings are commutative. Any unital quantale is an idempotent semiring under join and multiplication. Any bounded, distributive lattice is a commutative, idempotent semiring under join and meet.

In particular, a Boolean algebra is such a semiring. A Boolean ring is also a semiring indeed, a ring but it is not idempotent under addition. A Boolean semiring is a semiring isomorphic to a subsemiring of a Boolean algebra.

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## A Double Category Theoretic Analysis of Graded Linear Exponential Comonads

More generally, this likewise applies to the square matrices whose entries are elements of any other given semiring S, and this new semiring of matrices is generally non-commutative even though S may be commutative. The zero morphism and the identity are the respective neutral elements. N[x], polynomials with natural number coefficients form a commutative semiring. Tropical semirings are variously defined.

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## traduire de

Four equational axioms related to distributive law 3. We introduce the main subject of this study, R-graded linear exponential comonad. This concept first appeared in [ 3 , Definition 13] under the name exponential action. We adopt the following definition [ 7 , Sect. They satisfy four equational axioms in Fig. This graded linear exponential comonad is used to model the level of information flow [ 7 , Sect.