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## Monoids & Semigroups · aol/cyclops Wiki · GitHub

Received Feb 20; Accepted Oct 4. Introduction In mathematics, an ordered semigroup is a semigroup together with a partial order that is compatible with the semigroup operation. The converse of Proposition 4. Open in a separate window. Implication-based fuzzy interior ideals Fuzzy logic is an extension of set theoretic variables in terms of the linguistic variable truth. Acknowledgments The authors are very grateful to referees for their valuable comments and suggestions for improving this paper.

Contributor Information Asghar Khan, Email: moc. References 1. Semigroups characterized by their fuzzy bi-ideals J Fuzzy Math 10 2 — Bhakat SK. Bhakat SK, Das P. On the definition of a fuzzy subgroup Fuzzy Sets Syst 51 — Fuzzy subrings and ideals redefined Fuzzy Sets Syst 81 — Das PS. Fuzzy groups and level subgroups J Math Anal Appl 84 — Davvaz B.

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Davvaz B, Corsini P. Redfined fuzzy H v -submodules and many valued implications Inf Sci — Intuitionistic fuzzy H v -submodules Inf Sci — Davvaz B, Mahdavipour M. Roughness in modules Inf Sci — Generalized fuzzy interior ideals in semigroups Inf Sci — Redefined fuzzy implicative filters Inf Sci — Kehayopulu N. On left regular and left duo poe-semigroups Semigroup Forum 44 — On intra-regular ordered semigroups Semigroup Forum 46 — Khan A, Shabir M.

Kuroki N. Generalized fuzzy filters of R 0 -algebras Soft Comput 11 — Fuzzy semigroups, studies in Fuzziness and soft computing.

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Berlin: Springer; Murali V. Fuzzy points of equivalent fuzzy subsets Inf Sci — Shabir M, Khan A. Characterizations of ordered semigroups by the properties of their fuzzy generalized bi-ideals New Math Nat Comput 4 2 — Intuitionistic fuzzy interior ideals in ordered semigroups.

J Appl Math Inform. Shabir M, Khan A On fuzzy ordered semigroups to appear.

Comput Math Appl. Zadeh LA. Fuzzy sets Inf Control 8 — A new view of fuzzy hypernear-rings Inf Sci — Support Center Support Center.

For instance, the Monoid type class requires that combine be associative and empty be an identity element for combine. That means the following equalities should hold for any choice of x , y , and z. With these laws in place, functions parametrized over a Monoid can leverage them for say, performance reasons. A function that collapses a List[A] into a single A can do so with foldLeft or foldRight since combine is assumed to be associative, or it can break apart the list into smaller lists and collapse in parallel, such as.

Cats provides laws for type classes via the kernel-laws and laws modules which makes law checking type class instances easy. From cats-infographic by tpolecat.

## Monoids & Semigroups

Originally from alexknvl. Type classes Type classes are a powerful tool used in functional programming to enable ad-hoc polymorphism, more commonly known as overloading. Example: collapsing a list The following code snippets show code that sums a list of integers, concatenates a list of strings, and unions a list of sets.

The most important examples of bi-simple but not completely-simple semi-groups are: the bicyclic semi-groups and the four-spiral semi-group cf. Bicyclic semi-group ; [11]. The latter, , is given by generators and defining relations , , , , , , , , , ,.

It is isomorphic to a Rees semi-group of matrix type over a bicyclic semi-group with generators , where , with sandwich-matrix. In a sense, is minimal among the bi-simple not completely-simple semi-groups generated by a finite number of idempotents, and quite often it arises as a sub-semi-group of those semi-groups. Right simple semi-groups are also called semi-groups with right division, or semi-groups with right invertibility. The reason for this terminology is the following equivalent property of such semi-groups: For any elements there is an such that.

The right simple semi-groups containing idempotents are precisely the right groups cf.

## Special classes of semigroups

Right group. An important example of a right simple semi-group without idempotents is given by the semi-groups of all transformations of a set such that: 1 the kernel of is the equivalence relation on ; 2 the cardinality of the quotient set is ; 3 the set intersects each -class in at most one element; and 4 the set of -classes disjoint from has infinite cardinality , and. The semi-group is called a Teissier semi-group of type , and, if is the equality relation, it is called a Baer—Levi semi-group of type cf. A Teissier semi-group is an example of a right simple semi-group without idempotents that does not necessarily satisfy the right cancellation law.

Every right simple semi-group without idempotents can be imbedded in a suitable Teissier semi-group, while every such semi-group with the right cancellation law can be imbedded in a suitable Baer—Levi semi-group in both cases one can take. Various types of simple semi-groups often arise as "blocks" from which one can construct the semi-groups under consideration. For classical examples of simple semi-groups see Completely-simple semi-group ; Brandt semi-group ; Right group ; for bi-simple inverse semi-groups including structure theorems under certain restrictions on the semi-lattice of idempotents see [1] , [8] , [9].