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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. [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 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]

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]

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]

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]

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]

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]

We correct a mistake in the description of orthoalgebras as pastings of Boolean algebras. We present a corrected structural theorem. [ABSTRACT FROM AUTHOR]

We study the automorphism groups of the subalgebras T and T ( m) of the classical Toeplitz algebra which are invariant under a finite subgroup in S. We also give description for the automorphism group of the subalgebra of compact operators in Tm. The relationship between the automorphisms of T and T ( m) is revealed. [ABSTRACT FROM AUTHOR]

The notion of quasi-valuation maps based on a subalgebra and an ideal in BCK/BCI-algebras is introduced, and then several properties are investigated. Relations between a quasi-valuation map based on a subalgebra and a quasi-valuation map based on an ideal is established. In a BCI-algebra, a condition for a quasi-valuation map based on an ideal to be a quasi-valuation map based on a subalgebra is provided, and conditions for a real-valued function on a BCK/BCI-algebra to be a quasi-valuation map based on an ideal are discussed. Using the notion of a quasi-valuation map based on an ideal, (pseudo) metric spaces are constructed, and we show that the binary operation * in BCK-algebras is uniformly continuous. [ABSTRACT FROM AUTHOR]

An ordered algebra is called a sup-algebra if its underlying poset is a complete lattice and its operations are compatible with joins in each variable. In this article we study quotients and subalgebras of sup-algebras. We show that the congruence lattice of a sup-algebra is isomorphic to the lattice of its nuclei and dually isomorphic to the lattice of its meet-closed subalgebras. We also prove that the lattice of subalgebras of a sup-algebra is isomorphic to the lattice of its conuclei. [ABSTRACT FROM AUTHOR]

According to the orbit method, the construction of a unitary irreducible representation of a nilpotent Lie group requires a precise computation of some polarizing subalgebra subordinated to a linear functional in the linear dual of the corresponding Lie algebra. This important step is generally challenging from a computational viewpoint. In this paper, we provide an algorithmic approach to the construction of the well-known Vergne polarizing subalgebras. The algorithms presented in this paper are specifically designed so that they can be implemented in computer algebra systems. We also show there are instances where Vergne’s construction could be refined for the sake of efficiency. Finally, we adapt our refined procedure to free nilpotent finite-dimensional Lie algebras of step-two to obtain simple and precise descriptions of Vergne polarizing algebras corresponding to all linear functionals in a dense open subset of the linear dual of the corresponding Lie algebra. Also, a program written for Mathematica is presented at the end of the paper. [ABSTRACT FROM AUTHOR]

In the paper there are investigated some properties of Lie algebras, the construction which has a wide range of applications like computer sciences (especially to computer visions), geometry or physics, for example. We concentrate on the semidirect sum of algebras and there are extended some theoretic designs as conditions to be a center, a homomorphism or a derivative. The Killing form of the semidirect sum where the second component is an ideal of the first one is considered as well. [ABSTRACT FROM AUTHOR]

A sharp upper bound for the length of an algebra as a function of the index of nilpotency of its Jacobson radical and the length of the quotient algebra by the radical is obtained. [ABSTRACT FROM AUTHOR]

The study of SU-Algebra was initiated by Supawadee Keawrahun and Utsanee Leerawat. This paper introduces the notion of Bifuzzy SU-subalgebra and deals with some of their basic but interesting properties related to the Cartesian product and Homomorphism by applying the idea of a Bifuzzy subset. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that the weak subalgebra lattice uniquely determines the subalgebra lattice for locally finite algebras of a fixed finite type. However, this algebraic result turns out to be a very particular case of the following hypergraph result (which is interesting itself): A total directed hypergraph D of finite type is uniquely determined, in the class of all the directed hypergraphs of this type, by its skeleton up to the orientation of some pairwise edge-disjoint directed hypercycles and hyperpaths. The skeleton of D is a hypergraph obtained from D by omitting the orientation of all edges. [ABSTRACT FROM AUTHOR]

Abstract: We give two new criteria for a basic algebra to be biserial. The first one states that an algebra is biserial iff all subalgebras of the form eAe where e is supported by at most 4 vertices are biserial. The second one gives some condition on modules that must not exist for a biserial algebra. These modules have properties similar to the module with dimension vector for the path algebra of the quiver . Both criteria generalize criteria for an algebra to be Nakayama. They rely on the description of a basic biserial algebra in terms of quiver and relations given by R. Vila-Freyer and W. Crawley-Boevey . [Copyright &y& Elsevier]

We classify all unital subalgebras of the Cayley algebra 핆(q) over the finite field Fq, q = pn. We obtain the number of subalgebras of each type and prove that all isomorphic subalgebras are conjugate with respect to the automorphism group of 핆(q). We also determine the structure of the Moufang loops associated with each subalgebra of 핆(q). [ABSTRACT FROM AUTHOR]

We study the length of constructions such as a quotient algebra, a direct sum of algebras, and a subalgebra, which occur in the classical Birkhoff theorem. An upper bound for the length of local algebras is obtained. [ABSTRACT FROM AUTHOR]

The notion of bipolar fuzzy subalgebras and bipolar fuzzy ideals of a BCK/BCI-algebra is introduced, and several properties are investigated. Relations between a bipolar fuzzy subalgebra and a bipolar fuzzy ideal are given. A condition for bipolar fuzzy subalgebra to be a bipolar fuzzy ideal is provided, and the characterizations of a bipolar fuzzy ideal are stated. The concept of equivalence relations on the family of all bipolar fuzzy ideals of a BCK/BCI-algebra is considered, and some related properties are discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate two kinds of algebras of fuzzy sets, which are obtained by using Zadeh''s extension principle. We give conditions under which a homomorphism between two algebras induces a homomorphism between corresponding algebras of fuzzy sets. We prove that if the structure of truth values is a complete residuated lattice, the induced algebra of a subalgebra of an algebra can be embedded into the induced algebra of fuzzy sets of . For direct products we give conditions under which the direct product of algebras of fuzzy sets could be embedded into the algebra of fuzzy sets of the direct product. In the case of homomorphisms and direct products, the two kinds of algebras of fuzzy sets behave in different ways. [Copyright &y& Elsevier]

Abstract: We prove a few results on the sufficiency of generic and codimension-one fibre conditions for determination of the structure of algebras of transcendence degree one. We first show that over a Noetherian normal domain R, a faithfully flat subalgebra of a finitely generated algebra whose generic and codimension-one fibres are is necessarily the symmetric algebra of an invertible ideal of R. We next prove a structure theorem for a faithfully flat algebra over a locally factorial Krull domain R whose generic and codimension-one fibres are . For R local, we deduce a minimal sufficient condition for the algebra to be finitely generated and hence . [Copyright &y& Elsevier]

Let an observation X take finitely many values with probabilities p1(θ), … , pN(θ) depending on an abstract parameter θ ϵ Θ. It is proved that a statistic is a uniformly minimum variance unbiased estimator (UMVUE) if and only if it is measurable with respect to a subalgebra of the finite algebra generated by X. In general, this subalgebra is smaller than the minimal sufficient subalgebra for θ and is explicitly described. It is related to a special partition of a finite set of elements of an abstract linear space. [ABSTRACT FROM AUTHOR]

Abstract: A resultant-type identity for univariate polynomials is proved and used to characterise SAGBI bases of subalgebras generated by two polynomials. A new equivalent condition, expressed in terms of the degree of a field extension, for a pair of univariate polynomials to form a SAGBI basis is derived. [Copyright &y& Elsevier]

An Ockham algebra ( L; f) will be called minimal if it has exactly two subalgebras. In the generalised variety K, we describe the minimal Ockham algebras. [ABSTRACT FROM AUTHOR]

We prove that if A is an algebra that is supernilpotent with respect to the 2-term higher commutator, and B is a subalgebra of A , then B is representable as a retract of a finite subdirect power of A . [ABSTRACT FROM AUTHOR]