rss_2.0Discussiones Mathematicae - General Algebra and Applications FeedSciendo RSS Feed for Discussiones Mathematicae - General Algebra and Applicationshttps://sciendo.com/journal/DMGAAhttps://www.sciendo.comDiscussiones Mathematicae - General Algebra and Applications 's Coverhttps://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/6009d12199b8105a15d5b455/cover-image.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20220811T015640Z&X-Amz-SignedHeaders=host&X-Amz-Expires=604800&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20220811%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=8885c11f74951b2f4375aaab053f027b5fbcd7b19ab396cd216a7d3c626e2cbe200300On Partial Clones of -Termshttps://sciendo.com/article/10.7151/dmgaa.1376<abstract> <title style='display:none'>Abstract</title> <p>The main purpose of this paper is to generalize the concept of linear terms. A linear term is a term in which every variable occurs at most once. K. Denecke defined partial operations on linear terms and partial clones. Moreover, their properties are also studied. In the present paper, a generalized notion of the partial clone of linear terms, which is called <italic>k</italic>-terms clone, is presented and we also study its properties. We provide a characterization of the <italic>k</italic>-terms clone being free with respect to itself. Moreover, we attempt to define mappings analogue to the concept of hypersubstitutions.</p> </abstract>ARTICLE2021-09-06T00:00:00.000+00:00Fuzzy Distributive Pairs in Fuzzy Latticeshttps://sciendo.com/article/10.7151/dmgaa.1386<abstract> <title style='display:none'>Abstract</title> <p>We generalize the concept of a fuzzy distributive lattice by introducing the concepts of a fuzzy join-distributive pair and a fuzzy join-semidistributive pair in a fuzzy lattice. A relationship among a fuzzy join-distributive pair, a fuzzy join-semidistributive pair and a fuzzy join-modular pair is proved. It is shown that for a pair of fuzzy atoms, the notions of a fuzzy join-distributive pair and a fuzzy join-semidistributive pair coincide.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00Order of Finite Soft Quasigroups with Application to Egalitarianismhttps://sciendo.com/article/10.7151/dmgaa.1381<abstract> <title style='display:none'>Abstract</title> <p>In this work, a soft set (<italic>F</italic>, <italic>A</italic>) was introduced over a quasigroup (<italic>Q</italic>,) and the study of finite soft quasigroup was carried out, motivated by the study of algebraic structures of soft sets. By introducing the order of a finite soft quasigroup, various inequality relationships that exist between the order of a finite quasigroup, the order of its soft quasigroup and the cardinality of its set of parameters were established. By introducing the arithmetic mean π’œβ„±(<italic>F</italic>, <italic>A</italic>) and geometric mean 𝒒ℱ(<italic>F</italic>, <italic>A</italic>) of a finite soft quasigroup (<italic>F</italic>, <italic>A</italic>), a sort of Lagrange’s Formula |(<italic>F</italic>, <italic>A</italic>)| = |<italic>A</italic>|π’œβ„±(<italic>F</italic>, <italic>A</italic>) for finite soft quasigroup was gotten. Some of the inequalities gotten gave an upper bound for the order of a finite soft quasigroup in terms of the order of its quasigroup and cardinality of its set of parameters, and a lower bound for the order of the quasigroup in terms of the arithmetic mean of the finite soft quasigroup. A chain of inequalities called the Maclaurin’s inequality for any finite soft quasigroup (<italic>F</italic>, <italic>A</italic>)<sub>(</sub><italic><sub>Q</sub></italic><sub>,Β·)</sub> was shown to exist. A necessary and sufficient condition for a type of finite soft quasigroup to be extensible to a finite super soft quasigroup was established. This result is of practical use whenever a larger set of parameters is required. The results therein were illustrated with examples. Application to uniformity, equality and equity in distribution for social living is considered.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00On the Structure Space of a Π“ -Semigroup Via Its Left Operator Semigrouphttps://sciendo.com/article/10.7151/dmgaa.1379<abstract> <title style='display:none'>Abstract</title> <p>The structure space of a semigroup endowed with hull kernel topology is introduced and studied. Also the structure space of a Π“ -semigroup is defined and a homeomorphism has been established between structure space of a Π“ -semigroup and the structure space of its left operator semigroup. Moreover, various properties of structure space of a Π“ -semigroup are studied via its left operator semigroup.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00Stone Commutator Lattices and Baer Ringshttps://sciendo.com/article/10.7151/dmgaa.1380<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we transfer Daveyβ€˜s characterization for <italic>ΞΊ</italic> -Stone bounded distributive lattices to lattices with certain kinds of quotients, in particular to commutator lattices with certain properties, and obtain related results on prime, radical, complemented and compact elements, annihilators and congruences of these lattices. We then apply these results to certain congruence lattices, in particular to those of semiprime members of semi-degenerate congruence-modular varieties, and use this particular case to transfer Daveyβ€˜s Theorem to commutative unitary rings.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00New Kinds of Prefilters in EQ-Algebrashttps://sciendo.com/article/10.7151/dmgaa.1382<abstract> <title style='display:none'>Abstract</title> <p>In this paper, the notions of <italic>n</italic>-fold positive implicative prefilter and <italic>n</italic>-fold implicative prefilter in <italic>EQ</italic>-algebras are introduced and several properties, characterizations and equivalent conditions are provided. It is proved that the quotient EQ-algebra induced by an <italic>n</italic>-fold positive implicative prefilter is <italic>n</italic>-idempotent. Also, it is proved that in an <italic>n</italic>-idempotent EQ-algebra, any filter is an <italic>n</italic>-fold positive implicative filter. In the sequel, we investigate the relationships between these two types of prefilters. Finally, some characterizations of <italic>n</italic>-fold implicative prefilters in bounded EQ-algebras are given.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00Interior GE-Filters of GE-Algebrashttps://sciendo.com/article/10.7151/dmgaa.1385<abstract> <title style='display:none'>Abstract</title> <p>The notions of an interior GE-filter, a weak interior GE-filter and a belligerent interior GE-filter are introduced, and their relations and properties are investigated. Example of a GE-filter which is neither an interior GE-filter nor a weak interior GE-filter is provided. Relations between a weak interior GE-filter and an interior GE-filter are discussed, and conditions under which every weak interior GE-filter is an interior GE-filter are investigated. Relations between a belligerent interior GE-filter and an interior GE-filter are displayed, and conditions for an interior GE-filter to be a belligerent interior GE-filter are considered. Given a subset and an element, an interior GE-filter is established, and conditions for a subset to be a belligerent interior GE-filter are discussed. The extensibility of the beligerant interior GE-filter is debated. Relationships between weak interior GE-filter and belligerent interior GE-filter of type 1, type 2 and type 3 are founded.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00Generalized Centroid of Hyperringshttps://sciendo.com/article/10.7151/dmgaa.1387<abstract> <title style='display:none'>Abstract</title> <p>In this paper, the notion of generalized centroid is applied to hyperrings. We show that the generalized centroid <italic>C</italic> of a semiprime hyperring <italic>R</italic> is a regular hyperring. Also, we show that if <italic>C</italic> is a hyperfield, then <italic>R</italic> is a prime hyperring.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00Prime Ideals of Transitive -Algebrashttps://sciendo.com/article/10.7151/dmgaa.1383<abstract> <title style='display:none'>Abstract</title> <p>The notion of prime ideals is introduced in transitive <italic>BE</italic>-algebras. Prime ideals are characterized with the help of principal ideals. Prime ideal theorem is stated and derived for <italic>BE</italic>-algebras. The concept of minimal prime ideals is introduced in transitive <italic>BE</italic>-algebras. A decomposition theorem of proper ideals into minimal prime ideals is derived.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00Hyper Rl-Ideals in Hyper Residuated Latticeshttps://sciendo.com/article/10.7151/dmgaa.1377<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we introduce the notion of a (strong) hyper <italic>RL</italic>-ideal in hyper residuated lattices and give some properties and characterizations of them. Next, we characterize the (strong) hyper <italic>RL</italic>-ideals generated by a subset and give some characterizations of the lattice of these hyper <italic>RL</italic>-ideals. Particularly, we prove that this lattice is algebraic and compact elements are finitely generated hyper <italic>RL</italic>-ideals, and obtain some isomorphism theorems. Finally, we introduce the notion of nodal hyper <italic>RL</italic>-ideals in a hyper residuated lattice and investigate their properties. We prove that the set of nodal hyper <italic>RL</italic>-ideals is a complete Brouwerian lattice and under suitable operations is a Heyting algebra.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00Disjunctive Ideals of Almost Distributive Latticeshttps://sciendo.com/article/10.7151/dmgaa.1384<abstract> <title style='display:none'>Abstract</title> <p>The concept of disjunctive ideals is introduced in an Almost Distributive Lattice (ADL). It is proved that the set of all disjunctive ideals of an ADL forms a complete lattice. A necessary and sufficient condition is derived for an inverse homomorphic image of a disjunctive ideal of an ADL to be again a disjunctive ideal. Later, the concept of strongly disjunctive ideals is introduced in an ADL and their properties are studied. Some equivalent conditions are established for the set of all strongly disjunctive ideals to convert into a sublattice of the ideal lattice.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00Study of Additively Regular Π“ -Semirings and Derivationshttps://sciendo.com/article/10.7151/dmgaa.1378<abstract> <title style='display:none'>Abstract</title> <p>In this paper, the notions of commutator and derivation in additively regular -semirings with (<italic>A</italic><sub>2</sub>, Π“)-condition are introduced. We also characterize Jordan product for additively regular Π“ -semiring and establish some results which investigate the relationship between commutators, derivations and inner derivations. In 1957, E.C. Posner has shown that if there exists a non-zero centralizing derivation in a prime ring <italic>R</italic>, then <italic>R</italic> is commutative. This result is extended in the frame work of derivations of prime additively regular Π“ -semirings.</p> </abstract>ARTICLE2022-04-05T00:00:00.000+00:00A Result on Prime Rings with Generalized Derivationshttps://sciendo.com/article/10.7151/dmgaa.1373<abstract> <title style='display:none'>Abstract</title> <p>In this paper we investigate the following result. Let <italic>R</italic> be a prime ring, <italic>Q</italic> its symmetric Martindale quotient ring, <italic>C</italic> its extended centroid, <italic>I</italic> a nonzero ideal of <italic>R</italic>. If <italic>F</italic> and <italic>G</italic> are the two generalized derivation of <italic>R</italic> such that (F(xy) + G(yx))<italic><sup>n</sup></italic> βˆ’ (xy βˆ“ yx)<italic><sup>n</sup></italic> = 0, for all <italic>x, y</italic> ∈ <italic>I</italic>, then either <italic>R</italic> is commutative or <italic>F</italic> (<italic>x</italic>) = <italic>x</italic>, <italic>G</italic>(<italic>x</italic>) = βˆ“<italic>x</italic> for all <italic>x</italic> ∈ <italic>R</italic> and <italic>n</italic> = 1.</p> </abstract>ARTICLE2021-09-06T00:00:00.000+00:00Revisiting the Representation Theorem of Finite Distributive Lattices with Principal Congruences. A Approachhttps://sciendo.com/article/10.7151/dmgaa.1375<abstract> <title style='display:none'>Abstract</title> <p>A classical result of R.P. Dilworth states that every finite distributive lattice <italic>D</italic> can be represented as the congruence lattice of a finite lattice <italic>L</italic>. A sharper form was published in G. GrΓ€tzer and E.T. Schmidt in 1962, adding the requirement that all congruences in <italic>L</italic> be principal. Another variant, published in 1998 by the authors and E.T. Schmidt, constructs a planar semimodular lattice <italic>L</italic>. In this paper, we merge these two results: we construct <italic>L</italic> as a planar semimodular lattice in which all congruences are principal. This paper relies on the techniques developed by the authors and E.T. Schmidt in the 1998 paper.</p> </abstract>ARTICLE2021-09-06T00:00:00.000+00:00On Order Prime Divisor Graphs of Finite Groupshttps://sciendo.com/article/10.7151/dmgaa.1372<abstract> <title style='display:none'>Abstract</title> <p>The order prime divisor graph π’«π’Ÿ(<italic>G</italic>) of a finite group <italic>G</italic> is a simple graph whose vertex set is <italic>G</italic> and two vertices <italic>a, b</italic> ∈ <italic>G</italic> are adjacent if and only if either <italic>ab</italic> = <italic>e</italic> or <italic>o</italic>(<italic>ab</italic>) is some prime number, where <italic>e</italic> is the identity element of the group <italic>G</italic> and <italic>o</italic>(<italic>x</italic>) denotes the order of an element <italic>x</italic> ∈ <italic>G</italic>. In this paper, we establish the necessary and sufficient condition for the completeness of order prime divisor graph π’«π’Ÿ(<italic>G</italic>) of a group <italic>G</italic>. Concentrating on the graph π’«π’Ÿ(<italic>D<sub>n</sub></italic>), we investigate several properties like degrees, girth, regularity, Eulerianity, Hamiltonicity, planarity etc. We characterize some graph theoretic properties of π’«π’Ÿ (β„€<sub>n</sub>), π’«π’Ÿ (<italic>S<sub>n</sub></italic>), π’«π’Ÿ (<italic>A<sub>n</sub></italic>).</p> </abstract>ARTICLE2021-09-06T00:00:00.000+00:00On eGE-Algebrashttps://sciendo.com/article/10.7151/dmgaa.1362<abstract> <title style='display:none'>Abstract</title> <p>A new algebraic structure was introduced, called an eGE-algebra, which is a generalisation of a GE-algebra and investigated its properties. We explore the definition of filters and the quotient algebra associated with such filters.</p> </abstract>ARTICLE2021-09-06T00:00:00.000+00:00Classification of Elements in Elliptic Curve Over the Ring 𝔽[Ι›]https://sciendo.com/article/10.7151/dmgaa.1371<abstract> <title style='display:none'>Abstract</title> <p>Let 𝔽<italic><sub>q</sub></italic>[Ι›] := 𝔽<italic><sub>q</sub></italic> [<italic>X</italic>]/(<italic>X</italic><sup>4</sup> βˆ’ <italic>X</italic><sup>3</sup>) be a finite quotient ring where Ι›<sup>4</sup> = Ι›<sup>3</sup>, with 𝔽<italic><sub>q</sub></italic> is a finite field of order q such that q is a power of a prime number <italic>p</italic> greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽<italic><sub>q</sub></italic>[Ι›], Ι›<sup>4</sup> = Ι›<sup>3</sup> of characteristic p β‰  2, 3 given by homogeneous Weierstrass equation of the form <italic>Y</italic> <sup>2</sup><italic>Z</italic> = <italic>X</italic><sup>3</sup> + <italic>aXZ</italic><sup>2</sup> + <italic>bZ</italic><sup>3</sup> where a and b are parameters taken in 𝔽<italic><sub>q</sub></italic>[Ι›]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve <italic>E<sub>a,b</sub></italic>(𝔽<italic><sub>q</sub></italic>[Ι›]) and we will show that <italic>E<sub>Ο€</sub></italic><sub>0(</sub><italic><sub>a</sub></italic><sub>),</sub><italic><sub>Ο€</sub></italic><sub>0(</sub><italic><sub>b</sub></italic><sub>)</sub>(𝔽<italic><sub>q</sub></italic>) and <italic>E<sub>Ο€</sub></italic><sub>1(a)</sub>,<italic><sub>Ο€</sub></italic><sub>1(</sub><italic><sub>b</sub></italic><sub>)</sub>(𝔽<italic><sub>q</sub></italic>) are two elliptic curves over the finite field 𝔽<italic><sub>q</sub></italic>, such that <italic>Ο€</italic><sub>0</sub> is a canonical projection and <italic>Ο€</italic><sub>1</sub> is a sum projection of coordinate of element in 𝔽<italic><sub>q</sub></italic>[Ι›]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽<italic><sub>q</sub></italic>[Ι›].</p> </abstract>ARTICLE2021-09-06T00:00:00.000+00:00𝒩 -Prime Spectrum of Stone Almost Distributive Latticeshttps://sciendo.com/article/10.7151/dmgaa.1370<abstract> <title style='display:none'>Abstract</title> <p>Introduced the notions of annulets and 𝒩 -filters in stone Almost Distributive Lattices and investigated their properties. Utilized annulets to characterize the 𝒩 -filters. Derived that every proper 𝒩 -filter is the intersection of all 𝒩 -prime filters containing it and also proved that the set β„±<sub>𝒩</sub> (<italic>L</italic>) of all 𝒩 -filters is isomorphic to the class <italic>Con<sub>E</sub></italic>(<italic>L</italic>) of all <italic>G</italic>-extentions of <italic>L</italic>. Given some topological properties of the space of all 𝒩 -prime filters. Derived a necessary and sufficient condition for the space of all 𝒩 -prime filters to be a Hausdorff space.</p> </abstract>ARTICLE2021-09-06T00:00:00.000+00:00Green’s Relations on Submonoids of Generalized Hypersubstitutions of Type (n)https://sciendo.com/article/10.7151/dmgaa.1366<abstract> <title style='display:none'>Abstract</title> <p>A generalized hypersubstitution of type <italic>Ο„</italic> = (<italic>n</italic>) is a function which takes the <italic>n</italic>-ary operation symbol <italic>f</italic> to the term of the same type <italic>Οƒ</italic>(<italic>f</italic> ) which does not necessarily preserve the arity. Let <italic>Hyp<sub>G</sub></italic>(<italic>n</italic>) be the set of all these generalized hypersubstitutions of type (<italic>n</italic>). The set <italic>Hyp<sub>G</sub></italic>(<italic>n</italic>) with a binary operation and the identity generalized hypersubstitution forms a monoid. The objective of this paper is to study Green’s relations on the set of all regular elements of <italic>Hyp<sub>G</sub></italic>(<italic>n</italic>).</p> </abstract>ARTICLE2021-09-06T00:00:00.000+00:00On Partial Clones of -Termshttps://sciendo.com/article/10.7151/dmgaa.1367<abstract> <title style='display:none'>Abstract</title> <p>The main purpose of this paper is to generalize the concept of linear terms. A linear term is a term in which every variable occurs at most once. K. Denecke defined partial operations on linear terms and partial clones. Moreover, their properties are also studied. In the present paper, a generalized notion of the partial clone of linear terms, which is called <italic>k</italic>-terms clone, is presented and we also study its properties. We provide a characterization of the <italic>k</italic>-terms clone being free with respect to itself. Moreover, we attempt to define mappings analogue to the concept of hypersubstitutions.</p> </abstract>ARTICLE2021-09-06T00:00:00.000+00:00en-us-1