In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices.

The smallest semiring that is not a ring is the two-element Boolean algebra, e.g. with logical disjunction as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers under ordinary addition and multiplication, when including the number zero. Semirings are abundant, because a suitable multiplication operation arises as the function composition of endomorphism over any commutative monoid.

The theory of (associative) algebras over commutative rings can be generalized to one over commutative semirings.[citation needed]

Terminology

Some authors call semiring the structure without the requirement for there to be a or . 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.[1][a] This originated as a joke, suggesting that rigs are rings without negative elements. (And this is similar to using rng to mean a ring without a multiplicative identity.)

The term dioid (for "double monoid") has been used to mean semirings or other structures. It was used by Kuntzman in 1972 to denote a semiring.[2] (It is alternatively sometimes used for naturally ordered semirings[3] but the term was also used for idempotent subgroups by Baccelli et al. in 1992.[4])

Definition

A semiring is a set equipped with two binary operations and called addition and multiplication, such that:[5][6][7]

Explicitly stated, is a commutative monoid. Further, the following axioms tie to both operations:

Notation

The symbol is usually omitted from the notation; that is, is just written

Similarly, an order of operations is conventional, in which is applied before . That is, denotes .

For the purpose of disambiguation, one may write or to emphasize which structure the units at hand belong to.

If is an element of a semiring and , then -times repeated multiplication of with itself is denoted , and one similarly writes for the -times repeated addition.

Construction of new semirings

The zero ring with underlying set is also a semiring, called the trivial semiring. This triviality can be characterized via and so is often silently assumed as if it were an additional axiom. Now given any semiring, there are several ways to define new ones.

As noted, the natural numbers with its arithmetic structure form a semiring. The set equipped with the operations inherited from a semiring , is always a sub-semiring of .

If is a commutative monoid, function composition provides the multiplication to form a semiring: The set of endomorphisms forms a semiring, where addition is defined from pointwise addition in . The zero morphism and the identity are the respective neutral elements. If with a semiring, we obtain a semiring that can be associated with the square matrices with coefficients in , the matrix semiring using ordinary addition and multiplication rules of matrices. Yet more abstractly, given and a semiring, is always a semiring also. It is generally non-commutative even if was commutative.

Dorroh extensions: If is a semiring, then with pointwise addition and multiplication given by defines another semiring with mulitplicative unit . Very similarly, if is any sub-semiring of , one may also define a semiring on , just by replacing the repeated addition in the formula by multiplication. Indeed, these constructions even work under looser conditions, as the structure is not actually required to have a multiplicative unit.

Zerosumfree semirings are in a sense furthest away from being rings. Given a semiring, one may adjoin a new zero to the underlying set and thus obtain such a zerosumfree semiring that also lacks zero divisors. In particular, now and the old semiring is actually not a sub-semiring. One may then go on and adjoin new elements "on top" one at a time, while always respecting the zero. These two strategies also work under looser conditions. Sometimes the notations resp. are used when performing these constructions.

Adjoining a new zero to the trivial semiring, in this way, results in another semiring which may be expressed in terms of the logical connectives of disjunction and conjunction: . Consequently, this is the smallest semiring that is not a ring. Explicitly, it violates the ring axioms as for all , i.e. has no additive inverse. In the self-dual definition, the fault is with . (This is not to be conflated with the ring , whose addition functions as xor .) In the von Neumann model of the naturals, , and . The two-element semiring may be presented in terms of the set theoretic union and intersection as . Now this structure in fact still constitutes a semiring when is replaced by any inhabited set whatsoever.

The ideals on a semiring , with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring. The ideals of are in bijection with the ideals of . The collection of left ideals of (and likewise the right ideals) also have much of that algebraic structure, except that then does not function as a two-sided multiplicative identity.

If is a semiring and is an inhabited set, denotes the free monoid and the formal polynomials over its words form another semiring. For small sets, the generating elements are conventionally used to denote the polynomial semiring. For example, in case of a singleton such that , one writes . Zerosumfree sub-semirings of can be used to determine sub-semirings of .

Given a set , not necessarily just a singleton, adjoining a default element to the set underlying a semiring one may define the semiring of partial functions from to .

Given a derivation on a semiring , another the operation "" fulfilling can be defined as part of a new multiplication on , resulting in another semiring.

The above is by no means an exhaustive list of systematic constructions.

Derivations

Derivations on a semiring are the maps with and .

For example, if is the unit matrix and , then the subset of given by the matrices with is a semiring with derivation .

Properties

A basic property of semirings is that is not a left or right zero divisor, and that but also squares to itself, i.e. these have .

Some notable properties are inherited from the monoid structures: The monoid axioms demand unit existence, and so the set underlying a semiring cannot be empty. Also, the 2-ary predicate defined as , here defined for the addition operation, always constitutes the right canonical preorder relation. Reflexivity is witnessed by the identity. Further, is always valid, and so zero is the least element with respect to this preorder. Considering it for the commutative addition in particular, the distinction of "right" may be disregarded. In the non-negative integers , for example, this relation is anti-symmetric and strongly connected, and thus in fact a (non-strict) total order.

Below, more conditional properties are discussed.

Semifields

Any field is also a semifield, which in turn is a semiring in which also multiplicative inverses exist.

Rings

Any field is also a ring, which in turn is a semiring in which also additive inverses exist. Note that a semiring omits such a requirement, i.e., it requires only a commutative monoid, not a commutative group. The extra requirement for a ring itself already implies the existence of a multiplicative zero. This contrast is also why for the theory of semirings, the multiplicative zero must be specified explicitly.

Here , the additive inverse of , squares to . As additive differences always exist in a ring, is a trivial binary relation in a ring.

Commutative semirings

A semiring is called a commutative semiring if also the multiplication is commutative.[8] Its axioms can be stated concisely: It consists of two commutative monoids and on one set such that and .

The center of a semiring is a sub-semiring and being commutative is equivalent to being its own center.

The commutative semiring of natural numbers is the initial object among its kind, meaning there is a unique structure preserving map of into any commutative semiring.

The bounded distributive lattices are partially ordered, commutative semirings fulfilling certain algebraic equations relating to distributivity and idempotence. Thus so are their duals.

Ordered semirings

Notions or order can be defined using strict, non-strict or second-order formulations. Additional properties such as commutativity simplify the axioms.

Given a strict total order (also sometimes called linear order, or pseudo-order in a constructive formulation), then by definition, the positive and negative elements fulfill resp. . By irreflexivity of a strict order, if is a left zero divisor, then is false. The non-negative elements are characterized by , which is then written .

Generally, the strict total order can be negated to define an associated partial order. The asymmetry of the former manifests as . In fact in classical mathematics the latter is a (non-strict) total order and such that implies . Likewise, given any (non-strict) total order, its negation is irreflexive and transitive, and those two properties found together are sometimes called strict quasi-order. Classically this defines a strict total order – indeed strict total order and total order can there be defined in terms of one another.

Recall that "" defined above is trivial in any ring. The existence of rings that admit a non-trivial non-strict order shows that these need not necessarily coincide with "".

Additively idempotent semirings

A semiring in which every element is an additive idempotent, that is, for all elements , is called an (additively) idempotent semiring.[9] Establishing suffices. Be aware that sometimes this is just called idempotent semiring, regardless of rules for multiplication.

In such a semiring, is equivalent to and always constitutes a partial order, here now denoted . In particular, here . So additively idempotent semirings are zerosumfree and, indeed, the only additively idempotent semiring that has all additive inverses is the trivial ring and so this property is specific to semiring theory. Addition and multiplication respect the ordering in the sense that implies , and furthermore implies as well as , for all and .

If is addtively idempotent, then so are the polynomials in .

A semiring such that there is a lattice structure on its underlying set is lattice-ordered if the sum coincides with the meet, , and the product lies beneath the join . The lattice-ordered semiring of ideals on a semiring is not necessarily distributive with respect to the lattice structure.

More strictly than just additive idempotence, a semiring is called simple iff for all . Then also and for all . Here then functions akin to an additively infinite element. If is an additively idempotent semiring, then with the inherited operations is its simple sub-semiring. An example of an additively idempotent semiring that is not simple is the tropical semiring on with the 2-ary maximum function, with respect to the standard order, as addition. Its simple sub-semiring is trivial.

A c-semiring is an idempotent semiring and with addition defined over arbitrary sets.

An additively idempotent semiring with idempotent multiplication, , is called additively and multiplicatively idempotent semiring, but sometimes also just idempotent semiring. The commutative, simple semirings with that property are exactly the bounded distributive lattices with unique minimal and maximal element (which then are the units). Heyting algebras are such semirings and the Boolean algebras are a special case.

Further, given two bounded distributive lattices, there are constructions resulting in commutative additively-idempotent semirings, which are more complicated than just the direct sum of structures.

Number lines

In a model of the ring , one can define a non-trivial positivity predicate and a predicate as that constitutes a strict total order, which fulfills properties such as , or classically the law of trichotomy. With its standard addition and multiplication, this structure forms the strictly ordered field that is Dedekind-complete. By definition, all first-order properties proven in the theory of the reals are also provable in the decidable theory of the real closed field. For example, here is mutually exclusive with .

But beyond just ordered fields, the four properties listed below are also still valid in many sub-semirings of , including the rationals, the integers, as well as the non-negative parts of each of these structures. In particular, the non-negative reals, the non-negative rationals and the non-negative integers are such a semirings. The first two properties are analogous to the property valid in the idempotent semirings: Translation and scaling respect these ordered rings, in the sense that addition and multiplication in this ring validate

In particular, and so squaring of elements preserves positivity.

Take note of two more properties that are always valid in a ring. Firstly, trivially for any . In particular, the positive additive difference existence can be expressed as

Secondly, in the presence of a trichotomous order, the non-zero elements of the additive group are partitioned into positive and negative elements, with the inversion operation moving between them. With , all squares are proven non-negative. Consequently, non-trivial rings have a positive multiplicative unit,

Having discussed a strict order, it follows that and , etc.

Discretely ordered semirings

There are a few conflicting notions of discreteness in order theory. Given some strict order on a semiring, one such notion is given by being positive and covering , i.e. there being no element between the units, . Now in the present context, an order shall be called discrete if this is fulfilled and, furthermore, all elements of the semiring are non-negative, so that the semiring starts out with the units.

Denote by the theory of a commutative, discretly ordered semiring also validating the above four properties relating a strict order with the algebraic structure. All of its models have the model as its initial segment and Gödel incompleteness and Tarski undefinability already apply to . The non-negative elements of a commutative, discretely ordered ring always validate the axioms of . So a slightly more exotic model of the theory is given by the positive elements in the polynomial ring , with positivity predicate for defined in terms of the last non-zero coefficient, , and as above. While proves all -sentences that are true about , beyond this complexity one can find simple such statements that are independent of . For example, while -sentences true about are still true for the other model just defined, inspection of the polynomial demonstrates -independence of the -claim that all numbers are of the form or ("odd or even"). Showing that also can be discretely ordered demonstrates that the -claim for non-zero ("no rational squared equals ") is independent. Likewise, analysis for demonstrates independence of some statements about factorization true in . There are characterizations of primality that does not validate for the number .

In the other direction, from any model of one may construct an ordered ring, which then has elements that are negative with respect to the order, that is still discrete the sense that covers . To this end one defines an equivalence class of pairs from the original semiring. Roughly, the ring corresponds to the differences of elements in the old structure, generalizing the way in which the initial ring can be defined from . This, in effect, adds all the inverses and then the preorder is again trivial in that .

Beyond the size of the two-element algebra, no simple semiring starts out with the units. Being discretly ordered also stands in contrast to, e.g., the standard ordering on the semiring of non-negative rationals , which is dense between the units. For another example, can be ordered, but not discretely so.

Natural numbers

plus mathematical induction gives a theory equivalent to first-order Peano arithmetic . The theory is also famously not categorical, but is of course the intended model. proves that there are no zero divisors and it is zerosumfree and so no model of it is a ring.

The standard axiomatization of is more concise and the theory of its order is commonly treated in terms of the non-strict "". However, just removing the potent induction principle from that axiomatization does not leave a workable algebraic theory. Indeed, even Robinson arithmetic , which removes induction but adds back the predecessor existence postulate, does not prove the monoid axiom .

Complete semirings

A complete semiring is a semiring for which the additive monoid is a complete monoid, meaning that it has an infinitary sum operation for any index set and that the following (infinitary) distributive laws must hold:[10][11][12]

Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring.[13] For commutative, additively idempotent and simple semirings, this property is related to residuated lattices.

Continuous semirings

A continuous semiring is similarly defined as one for which the addition monoid is a continuous monoid. That is, partially ordered with the least upper bound property, and for which addition and multiplication respect order and suprema. The semiring with usual addition, multiplication and order extended is a continuous semiring.[14]

Any continuous semiring is complete:[10] this may be taken as part of the definition.[13]

Star semirings

A star semiring (sometimes spelled starsemiring) is a semiring with an additional unary operator ,[9][11][15][16] satisfying

A Kleene algebra is a star semiring with idempotent addition and some additional axioms. They are important in the theory of formal languages and regular expressions.[11]

Complete star semirings

In a complete star semiring, the star operator behaves more like the usual Kleene star: for a complete semiring we use the infinitary sum operator to give the usual definition of the Kleene star:[11]

where

Note that star semirings are not related to *-algebra, where the star operation should instead be thought of as complex conjugation.

Conway semiring

A Conway semiring is a star semiring satisfying the sum-star and product-star equations:[9][17]

Every complete star semiring is also a Conway semiring,[18] but the converse does not hold. An example of Conway semiring that is not complete is the set of extended non-negative rational numbers with the usual addition and multiplication (this is a modification of the example with extended non-negative reals given in this section by eliminating irrational numbers).[11] An iteration semiring is a Conway semiring satisfying the Conway group axioms,[9] associated by John Conway to groups in star-semirings.[19]

Examples

Similarly, in the presence of an appropriate order with bottom element,

Note that . More regarding additively idempotent semirings,

More using monoids,

Regarding sets and similar abstractions,

Semiring of sets

See also: Ring of sets § semiring

A semiring (of sets)[28] is a (non-empty) collection of subsets of such that

    • If (3) holds, then if and only if
  1. If then
  2. If then there exists a finite number of mutually disjoint sets such that

Conditions (2) and (3) together with imply that Such semirings are used in measure theory. An example of a semiring of sets is the collection of half-open, half-closed real intervals

A semialgebra[29] or elementary family[30] is a collection of subsets of satisfying the semiring properties except with (3) replaced with:

This condition is stronger than (3), which can be seen as follows. If is a semialgebra and , then we can write for disjoint . Then:

and every since it is closed under intersection, and disjoint since they are contained in the disjoint 's. Moreover, the condition is strictly stronger: any that is both a ring and a semialgebra is an algebra, hence any ring that is not an algebra is also not a semialgebra (e.g. the collection of finite sets on an infinite set ).

Star semirings

Several structures mentioned above can be equipped with a star operation.

Applications

The and tropical semirings on the reals are often used in performance evaluation on discrete event systems. The real numbers then are the "costs" or "arrival time"; the "max" operation corresponds to having to wait for all prerequisites of an events (thus taking the maximal time) while the "min" operation corresponds to being able to choose the best, less costly choice; and + corresponds to accumulation along the same path.

The Floyd–Warshall algorithm for shortest paths can thus be reformulated as a computation over a algebra. Similarly, the Viterbi algorithm for finding the most probable state sequence corresponding to an observation sequence in a hidden Markov model can also be formulated as a computation over a algebra on probabilities. 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.[31][32]

Generalizations

A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a semigroup rather than a monoid. Such structures are called hemirings[33] or pre-semirings.[34] A further generalization are left-pre-semirings,[35] which additionally do not require right-distributivity (or right-pre-semirings, which do not require left-distributivity).

Yet a further generalization are near-semirings: in addition to not requiring a neutral element for product, or right-distributivity (or left-distributivity), they do not require addition to be commutative. Just as cardinal numbers form a (class) semiring, so do ordinal numbers form a near-semiring, when the standard ordinal addition and multiplication are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead.

In category theory, a 2-rig is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig.

See also

Notes

  1. ^ For an example see the definition of rig on Proofwiki.org
  2. ^ a b This is a complete star semiring and thus also a Conway semiring.[11]

Citations

  1. ^ Głazek (2002), p. 7
  2. ^ Kuntzmann, J. (1972). Théorie des réseaux (graphes) (in French). Paris: Dunod. Zbl 0239.05101.
  3. ^ Semirings for breakfast, slide 17
  4. ^ Baccelli, François Louis; Olsder, Geert Jan; Quadrat, Jean-Pierre; Cohen, Guy (1992). Synchronization and linearity. An algebra for discrete event systems. Wiley Series on Probability and Mathematical Statistics. Chichester: Wiley. Zbl 0824.93003.
  5. ^ Berstel & Perrin (1985), p. 26
  6. ^ a b c d e Lothaire (2005), p. 211
  7. ^ Sakarovitch (2009), pp. 27–28
  8. ^ Lothaire (2005), p. 212
  9. ^ a b c d e Ésik, Zoltán (2008). "Iteration semirings". In Ito, Masami (ed.). Developments in language theory. 12th international conference, DLT 2008, Kyoto, Japan, September 16–19, 2008. Proceedings. Lecture Notes in Computer Science. Vol. 5257. Berlin: Springer-Verlag. pp. 1–20. doi:10.1007/978-3-540-85780-8_1. ISBN 978-3-540-85779-2. Zbl 1161.68598.
  10. ^ a b c Kuich, Werner (2011). "Algebraic systems and pushdown automata". In Kuich, Werner (ed.). Algebraic foundations in computer science. Essays dedicated to Symeon Bozapalidis on the occasion of his retirement. Lecture Notes in Computer Science. Vol. 7020. Berlin: Springer-Verlag. pp. 228–256. ISBN 978-3-642-24896-2. Zbl 1251.68135.
  11. ^ a b c d e f g h i j k l m n o Droste & Kuich (2009), pp. 7–10
  12. ^ Kuich, Werner (1990). "ω-continuous semirings, algebraic systems and pushdown automata". In Paterson, Michael S. (ed.). Automata, Languages and Programming: 17th International Colloquium, Warwick University, England, July 16–20, 1990, Proceedings. Lecture Notes in Computer Science. Vol. 443. Springer-Verlag. pp. 103–110. ISBN 3-540-52826-1.
  13. ^ a b Sakaraovich (2009), p. 471
  14. ^ Ésik, Zoltán; Leiß, Hans (2002). "Greibach normal form in algebraically complete semirings". In Bradfield, Julian (ed.). Computer science logic. 16th international workshop, CSL 2002, 11th annual conference of the EACSL, Edinburgh, Scotland, September 22–25, 2002. Proceedings. Lecture Notes in Computer Science. Vol. 2471. Berlin: Springer-Verlag. pp. 135–150. Zbl 1020.68056.
  15. ^ Lehmann, Daniel J. (1977), "Algebraic structures for transitive closure" (PDF), Theoretical Computer Science, 4 (1): 59–76, doi:10.1016/0304-3975(77)90056-1
  16. ^ Berstel & Reutenauer (2011), p. 27
  17. ^ Ésik, Zoltán; Kuich, Werner (2004). "Equational axioms for a theory of automata". In Martín-Vide, Carlos (ed.). Formal languages and applications. Studies in Fuzziness and Soft Computing. Vol. 148. Berlin: Springer-Verlag. pp. 183–196. ISBN 3-540-20907-7. Zbl 1088.68117.
  18. ^ Droste & Kuich (2009), p. 15, Theorem 3.4
  19. ^ Conway, J.H. (1971). Regular algebra and finite machines. London: Chapman and Hall. ISBN 0-412-10620-5. Zbl 0231.94041.
  20. ^ a b c Guterman, Alexander E. (2008). "Rank and determinant functions for matrices over semirings". In Young, Nicholas; Choi, Yemon (eds.). Surveys in Contemporary Mathematics. London Mathematical Society Lecture Note Series. Vol. 347. Cambridge University Press. pp. 1–33. ISBN 978-0-521-70564-6. ISSN 0076-0552. Zbl 1181.16042.
  21. ^ a b c Sakarovitch (2009), p. 28.
  22. ^ a b c Berstel & Reutenauer (2011), p. 4
  23. ^ Speyer, David; Sturmfels, Bernd (2009) [2004]. "Tropical Mathematics". Math. Mag. 82 (3): 163–173. arXiv:math/0408099. doi:10.4169/193009809x468760. S2CID 119142649. Zbl 1227.14051.
  24. ^ Speyer, David; Sturmfels, Bernd (2009). "Tropical Mathematics". Mathematics Magazine. 82 (3): 163–173. doi:10.1080/0025570X.2009.11953615. ISSN 0025-570X. S2CID 15278805.
  25. ^ John C. Baez (6 Nov 2001). "quantum mechanics over a commutative rig". Newsgroupsci.physics.research. Usenet: 9s87n0$iv5@gap.cco.caltech.edu. Retrieved November 25, 2018.
  26. ^ Bard, Gregory V. (2009), Algebraic Cryptanalysis, Springer, Section 4.2.1, "Combinatorial Classes", ff., pp. 30–34, ISBN 9780387887579
  27. ^ Schanuel S.H. (1991) Negative sets have Euler characteristic and dimension. In: Carboni A., Pedicchio M.C., Rosolini G. (eds) Category Theory. Lecture Notes in Mathematics, vol 1488. Springer, Berlin, Heidelberg
  28. ^ Noel Vaillant, Caratheodory's Extension, on probability.net
  29. ^ Durrett (2019), pp. 3–4
  30. ^ Folland (1999), p. 23
  31. ^ Pair (1967), p. 271.
  32. ^ Derniame & Pair (1971)
  33. ^ Golan (1999), p. 1, Ch 1
  34. ^ Gondran & Minoux (2008), p. 22, Ch 1, §4.2.
  35. ^ Gondran & Minoux (2008), p. 20, Ch 1, §4.1.

Bibliography

Further reading