On smallest (generalized) ideals and semilattices of (2,2)-regular non-associative ordered semigroups

An ordered AG-groupoid can be referred to as a non-associativeordered semigroup, as the main di¤erence between an ordered semigroup and anordered AG-groupoid is the switching of an associative law. In this paper, wede ne the smallest left (right) ideals in an ordered AG-groupoid and use them tocharacterize a (2; 2)-regular class of a unitary ordered AG-groupoid along with itssemilattices and (2 ;2 _q)-fuzzy left (right) ideals. We also give the conceptof an ordered A*G**-groupoid and investigate its structural properties by usingthe generated ideals and (2 ;2 _q)-fuzzy left (right) ideals. These concepts willverify the existing characterizations and will help in achieving more generalizedresults in future works.

Partial Menger algebras of terms

The superposition operation [Formula: see text] [Formula: see text] [Formula: see text], maps to each [Formula: see text]-tuple of [Formula: see text]-ary operations on a set [Formula: see text] an [Formula: see text]-ary operation on [Formula: see text] and satisfies the so-called superassociative law, a generalization of the associative law. The corresponding algebraic structures are Menger algebras of rank [Formula: see text]. A partial algebra of type [Formula: see text] which satisfies the superassociative law as weak identity is said to be a partial Menger algebra of rank [Formula: see text]. As a generalization of linear terms we define [Formula: see text]-terms as terms where each variable occurs at most [Formula: see text]-times. It will be proved that [Formula: see text]-ary [Formula: see text]-terms form partial Menger algebras of rank [Formula: see text]. In this paper, some algebraic properties of partial Menger algebras such as generating systems, homomorphic images and freeness are investigated. As generalization of hypersubstitutions and linear hypersubstitutions we consider [Formula: see text]-hypersubstitutions.

Partial clones

A set [Formula: see text] of operations defined on a nonempty set [Formula: see text] is said to be a clone if [Formula: see text] is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the [Formula: see text]-ary operations defined on set [Formula: see text] for all natural numbers [Formula: see text] and the operations are the so-called superposition operations [Formula: see text] for natural numbers [Formula: see text] and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set [Formula: see text] and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.

A Kind of Variation Symmetry: Tarski Associative Groupoids (TA-Groupoids) and Tarski Associative Neutrosophic Extended Triplet Groupoids (TA-NET-Groupoids)

The associative law reflects symmetry of operation, and other various variation associative laws reflect some generalized symmetries. In this paper, based on numerous literature and related topics such as function equation, non-associative groupoid and non-associative ring, we have introduced a new concept of Tarski associative groupoid (or transposition associative groupoid (TA-groupoid)), presented extensive examples, obtained basic properties and structural characteristics, and discussed the relationships among few non-associative groupoids. Moreover, we proposed a new concept of Tarski associative neutrosophic extended triplet groupoid (TA-NET-groupoid) and analyzed related properties. Finally, the following important result is proved: every TA-NET-groupoid is a disjoint union of some groups which are its subgroups.

A Study of Ordered Ag-Groupoids in terms of Semilattices via Smallest (Fuzzy) Ideals

An ordered AG-groupoid can be referred to as an ordered left almost semigroup, as the main difference between an ordered semigroup and an ordered AG-groupoid is the switching of an associative law. In this paper, we define the smallest one-sided ideals in an ordered AG-groupoid and use them to characterize a strongly regular class of a unitary ordered AG-groupoid along with its semilattices and fuzzy one-sided ideals. We also introduce the concept of an orderedAG⁎⁎⁎-groupoid and investigate its structural properties by using the generated ideals and fuzzy one-sided ideals. These concepts will verify the existing characterizations and will help in achieving more generalized results in future works.

On Characterization of Open Sets in Euclidean Spaces

A ternary semigroup is a nonempty set T together with a ternary oper- ation [abc] satisfying the associative law [[abc] de] = [a [bcd] e] = [ab [cde]] for all a, b, c, d, e ε T. A map f between topological spaces X and Y is called open if the image of each set open in X is open in Y. The pur- pose of this paper is to give an abstract characterization of the ternary semigroups of open maps defined on open sets in Euclidean n-spaces.

2. The laws of algebra

‘The laws of algebra’ explores the three laws that govern arithmetic operations and explains how these rules are extended so that they continue to be respected as we pass from one number system to a greater one that subsumes the former. The associative law of addition shows that that (a + b) + c = a + (b + c), and the associative law of multiplication is a(bc) = (ab)c. The distributive law tells us how to multiply out the brackets: a(b + c) = ab + ac. The commutative law of addition is a + b = b + a, a law that holds equally well for multiplication: ab = ba.

GELANGGANG ARTIN

A nonempty set R is said to be a ring if we can dene two binary operationsin R, denoted by + and respectively, such that for all a; b; c 2 R, R is an Abelian groupunder addition, closed under multiplication, and satisfy the associative law under multi-plication and distributive law. Let R be a ring. R is an Artin ring if every nonempty setof ideals has the minimal element. In this paper, the Artin ring and some characteristicsof it will be discussed.