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For an arbitrary unitary octonion algebra, we determine all subalgebras. It turns out that every subalgebra of dimension less than four is associative, while every subalgebra of dimension greater than four is not associative. In any split octonion algebra, there are both associative and non-associative subalgebras of dimension four. Except for one-dimensional subalgebras spanned by idempotents, any two isomorphic subalgebras are in the same orbit under automorphisms. [ABSTRACT FROM AUTHOR]

We introduce the concept of neutrosophic soft Lie subalgebras of a Lie algebra and investigate some of their properties. The Cartesian product of neutrosophic soft Lie subalgebras will be discussed. In particular, the homomorphisms of neutrosophic soft Lie algebras is introduced and investigated some of their properties.

We introduce the concept of neutrosophic soft Lie subalgebras of a Lie algebra and investigate some of their properties. The Cartesian product of neutrosophic soft Lie subalgebras will be discussed. In particular, the homomorphisms of neutrosophic soft Lie algebras is introduced and investigated some of their properties. [ABSTRACT FROM AUTHOR]

Ideals in BCK/BCI algebra based on YεJ -fuzzy sets are studied. The fundamental properties of the level set of YεJ -fuzzy sets are investigate first. The concept of (closed) YεJ -fuzzy ideals in BCK/BCI-algebras is introduces, and several properties are investigated. The relationship between YεJ -fuzzy ideal and YεJ -fuzzy subalgebra are discussed, and also the relationship between YεJ -fuzzy ideal and fuzzy ideal is identified. The characterization of (closed) YεJ -fuzzy ideal using the Y-level set is established. The necessary and sufficient conditions for YεJ -fuzzy ideal to be closed is explored, and conditions for YεJ -fuzzy subalgebra to be YεJ -fuzzy ideal are provided. [ABSTRACT FROM AUTHOR]

In this paper, we present a new multidimensional public-key cryptosystem that is commutative and associative based on the subalgebra of Quintuple algebra called SQN which utilizes a novel mathematical structure of two public keys and four private keys. This new structure has made the public key system more secure. [ABSTRACT FROM AUTHOR]

In the present paper, we examined the extendibility of evolution subalgebras generated by idempotents of evolution algebras. The extendibility of the isomorphism of such subalgebras to the entire algebra was investigated. Moreover, the existence of an evolution algebra generated by arbitrary idempotents was also studied. Furthermore, we described the tensor product of algebras generated by arbitrary idempotents and found the conditions of the tensor decomposability of four-dimensional S-evolution algebras. This paper's findings shed light on the field of algebraic structures, particularly in studying evolution algebras. By examining the extendibility of evolution subalgebras generated by idempotents, we provide insights into the structural properties and relationships within these algebras. Understanding the isomorphism of such subalgebras and their extension allows a deeper comprehension of the overall algebraic structure and its behaviour. [ABSTRACT FROM AUTHOR]

In this study, a neutrosophic N -subalgebra and neutrosophic N-ideal of a Sheffer stroke BCK-algebras are defined. It is shown that the level-set of a neutrosophic N-subalgebra (ideal) of a Sheffer stroke BCK-algebra is a subalgebra (ideal) of this algebra and vice versa. Then we present that the family of all neutrosophic N-subalgebras of a Sheffer stroke BCK-algebra forms a complete distributive modular lattice and that every neutrosophic N-ideal of a Sheffer stroke BCK-algebra is the neutrosophic N -subalgebra but the inverse does not usually hold. Also, relationships between neutrosophic N-ideals of Sheffer stroke BCK-algebras and homomorphisms are analyzed. Finally, we determine special subsets of a Sheffer stroke BCK-algebra by means of N-functions on this algebraic structure and examine the cases in which these subsets are its ideals. [ABSTRACT FROM AUTHOR]

The aim of the study is to introduce a neutrosophic N −subalgebra and neutrosophic N −ideal of a Sheffer stroke BCH-algebras. We prove that the level-set of a neutrosophic N −subalgebra (neutrosophic N −ideal) of a Sheffer stroke BCH-algebra is its subalgebra (ideal) and vice versa. Then it is shown that the family of all neutrosophic N −subalgebras of a Sheffer stroke BCH-algebra forms a complete distributive modular lattice. Also, we state that every neutrosophic N −ideal of a Sheffer stroke BCH-algebra is its neutrosophic N −subalgebra but the inverse is generally not true. We examine relationships between neutrosophic N −ideals of Sheffer stroke BCH-algebras by means of a surjective homomorphism between these algebras. Finally, certain subsets of a Sheffer stroke BCH-algebra are defined by means of N −functions on this algebraic structure and some properties are investigated.

In this paper, we introduce the concepts of fuzzy closed ideals of a bounded BE-algebra and prove basic properties. Using the idea of fuzzy closed ideals of a bounded BE-algebras, we discuss some related properties. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of subalgebra and discusses some properties of subalgebra. We also examine different types of ideals and how they behave. We also describe some properties of a homomorphism. Finally, we present quotient algebra and discuss isomorphism theorems. [ABSTRACT FROM AUTHOR]

In the present paper, we examined the extendibility of evolution subalgebras generated by idempotents of evolution algebras. The extendibility of the isomorphism of such subalgebras to the entire algebra was investigated. Moreover, the existence of an evolution algebra generated by arbitrary idempotents was also studied. Furthermore, we described the tensor product of algebras generated by arbitrary idempotents and found the conditions of the tensor decomposability of four-dimensional S-evolution algebras. This paper’s findings shed light on the field of algebraic structures, particularly in studying evolution algebras. By examining the extendibility of evolution subalgebras generated by idempotents, we provide insights into the structural properties and relationships within these algebras. Understanding the isomorphism of such subalgebras and their extension allows a deeper comprehension of the overall algebraic structure and its behaviour.

The aim of the study is to introduce a neutrosophic N--subalgebra and neutrosophic N--ideal of a Sheffer stroke BCH-algebras. We prove that the level-set of a neutrosophic N--subalgebra (neutrosophic N--ideal) of a Sheffer stroke BCH-algebra is its subalgebra (ideal) and vice versa. Then it is shown that the family of all neutrosophic N--subalgebras of a Sheffer stroke BCH-algebra forms a complete distributive modular lattice. Also, we state that every neutrosophic N--ideal of a Sheffer stroke BCH-algebra is its neutrosophic N--subalgebra but the inverse is generally not true. We examine relationships between neutrosophic N--ideals of Sheffer stroke BCH-algebras by means of a surjective homomorphism between these algebras. Finally, certain subsets of a Sheffer stroke BCH-algebra are defined by means of N--functions on this algebraic structure and some properties are investigated. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Complex neutrosophic Lie subalgebras and complex neutrosophic ideals of Lie algebras are defined in this paper. Each component in complex neutrosophic Lie algebra has magnitude and phase terms. Some characteristics of complex neutrosophic Lie subalgebras (ideals) and some of their operations like intersection and Cartesian product are also discussed. Moreover, the relationship between complex neutrosophic Lie subalgebras (ideals) and neutrosophic Lie subalgebras (ideals) is investigated. Finally, the image and the inverse image of complex neutrosophic Lie subalgebra under Lie algebra homomorphisms are defined and the properties of complex neutrosophic Lie subalgebras and complex neutrosophic ideals under homomorphisms of Lie algebras are studied. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Dalian University of Technology / Dalian Ligong Daxue Xuebao is the property of Journal of Dalian University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we first introduce some subalgebras in a reduct of the truth value algebra of type-2 fuzzy sets and prove one of them is a bisemilattice. Then we also obtain three necessary and sufficient conditions under which subalgebras of the truth value algebra of type-2 fuzzy sets form a bisemilattice. Next we investigate some subalgebras of this kind of bisemilattice and their characterizations. Finally, we introduce some subalgebras different from the bisemilattice and show a necessary and sufficient condition that a function is convex. [ABSTRACT FROM AUTHOR]

Let be the semifield with zero of nonnegative real numbers or the semifield of positive real numbers. The set of all continuous functions with pointwise operations of addition and multiplication of functions defined on an arbitrary topological space forms the semiring By a subalgebra we mean a nonempty subset of such that for any and any For arbitrary topological spaces and we describe isomorphisms of the semirings and and isomorphisms of the lattices of their subalgebras (subalgebras with unity). [ABSTRACT FROM AUTHOR]

All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classifications of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of isomorphism classes of all simple two-dimensional algebras is given. In the study of ideals and subalgebras, the number of them depends on roots of certain system of polynomials at structure constants of the algebra. We also give explicit forms of the polynomials. [ABSTRACT FROM AUTHOR]

This paper attempts to generalize universal algebras on classical sets to L-sets when L is a GL-quantale. Some basic notions of fuzzy universal algebra on an L-set are introduced, such as subalgebra, quotient algebra, homomorphism, congruence, and direct product etc. The properties of them are studied. L-valued power algebra is also introduced and it is shown there is an onto homomorphism from P(A)/R+ to P(A/R) for any congruence R on L-set A. [ABSTRACT FROM AUTHOR]

We consider the lattice of subalgebras of a semifield U(X) of positive continuous functions on an arbitrary topological space X and its sublattice of subalgebras with unity. We prove that each isomorphism of the lattices of subalgebras with unity of semifields U(X) and U(Y) is induced by a unique isomorphism of the semifields. The same result holds for lattices of all subalgebras excluding the case of the double-point Tychonoff extension of spaces. [ABSTRACT FROM AUTHOR]

The work on the theory of Groebner bases for ideals in a polynomial ring with countably infinite indeterminates over a field [5] has created impetus to develop the theory of Sagbi bases [6] and Sagbi Groebner bases [3] in the same polynomial ring. This paper demonstrates the construction of Sagbi basis and Sagbi Groebner basis using the technique of constructing these bases in a polynomial ring with finite indeterminates. [ABSTRACT FROM AUTHOR]

If an algebra satisfies the polynomial identity (for short, is ), then is trivially Lie solvable of index (for short, is ). We prove that the converse holds for subalgebras of the upper triangular matrix algebra any commutative ring, and . We also prove that if a ring S is (respectively, ), then the subring of comprising the upper triangular matrices with constant main diagonal, is (respectively, ) for all . We also study two related questions, namely whether, for a field F, an subalgebra of for some n, with (F-)dimension larger than the maximum dimension of a subalgebra of , exists, and whether a subalgebra of with (the mentioned) maximum dimension, other than the typical subalgebras of with maximum dimension, which were described by Domokos and refined by van Wyk and Ziembowski, exists. Partial results with regard to these two questions are obtained. [ABSTRACT FROM AUTHOR]

The paper is devoted to introduce the notion of rough sets within the context of residuated lattices. By considering the notion of a residuated lattice, we concern a relationship between rough sets theory and residuated lattices theory. We shall introduce the notion of rough subalgebra (resp. filter) with respect to a filter of a residuated lattice, which is an extended notion of subalgebra (resp. filter) in a residuated lattice and investigate some of their properties. [ABSTRACT FROM AUTHOR]

Let L 1 and L 2 be two finite dimensional Lie algebras on arbitrary field F with no common direct factor and L = L 1 ⊕ L 2 . In this article, we express the structure and dimension of derivation algebra of L , D e r (L) , and some of their subalgebras in terms of D e r (L 1) , D e r (L 2) , H o m (L 1 , Z (L 2)) , and H o m (L 2 , Z (L 1)). [ABSTRACT FROM AUTHOR]

Denote by R + ∨ the semifield with zero of nonnegative real numbers with operations of max-addition and multiplication. Let X be a topological space and C∨(X) be the semiring of continuous nonnegative functions on X with pointwise operation max-addition and multiplication of functions. By a subalgebra we mean a nonempty subset A of C∨(X) such that f ∨ g, fg, rf ∈ A for any f, g ∈ A, r ∈ R + ∨ . We consider the lattice A (C∨(X)) of subalgebras of the semiring C∨(X) and its sublattice A 1 (C∨(X)) of subalgebras with unity. The main result of the paper is the proof of the definability of the semiring C∨(X) both by the lattice A (C∨(X)) and by its sublattice A 1 (C∨(X)). [ABSTRACT FROM AUTHOR]

Let Mold n , d be the moduli of rank d subalgebras of M n over Z. For x ∈ Mold n , d , let A (x) ⊆ M n (k (x)) be the subalgebra of M n corresponding to x , where k (x) is the residue field of x. In this article, we apply Hochschild cohomology to Mold n , d. The dimension of the tangent space T Mold n , d / Z , x of Mold n , d over Z at x can be calculated by the Hochschild cohomology H 1 (A (x) , M n (k (x)) / A (x)). We show that H 2 (A (x) , M n (k (x)) / A (x)) = 0 is a sufficient condition for the canonical morphism Mold n , d → Z being smooth at x. We also calculate H i (A , M n (k) / A) for several R -subalgebras A of M n (R) over a commutative ring R. In particular, we summarize the results on H i (A , M n (k) / A) for all k -subalgebras A of M n (k) over an algebraically closed field k in the case n = 2 , 3. [ABSTRACT FROM AUTHOR]

Based on the notion of Q-sup-lattices (a fuzzy counterpart of complete join-semilattices valuated in a commutative quantale), we present the concept of Q-sup-algebras { Q-sup-lattices endowed with a collection of fini- tary operations compatible with the fuzzy joins. Similarly to the crisp case investigated in [30], we characterize their subalgebras and quotients, and fol- lowing [20], we show that the category of Q-sup-algebras is isomorphic to a certain subcategory of a category of Q-modules. [ABSTRACT FROM AUTHOR]