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We study the categories of matrix representations of nilpotent semigroups over an arbitrary field. Our main attention is given to the Auslander matrix algebras as one of possible forms of specifying the categories of representations, which have finitely many equivalence classes of indecomposable objects, and to the analysis of their discrete characteristics. The concept of Auslander algebra is generalized to the category of representations without auxiliary conditions. For any nilpotent cyclic semigroup, we describe its Auslander algebra and determine its Σ -function. [ABSTRACT FROM AUTHOR]

We study stability of the three geometric properties: uniform convexity, nearly uniform convexity, and property (β)$(\beta)$ under the Yoshikawa–Sparr interpolation method when the resulting interpolation space is considered with various equivalent norms. We give an example which shows that interpolation spaces obtained by the discrete and continuous versions of the method need not be isometric and present a method of transferring geometric properties from the discrete case to the continuous one. [ABSTRACT FROM AUTHOR]

Leibniz algebras are non-antisymmetric generalizations of Lie algebras. In this paper, we investigate the properties of complete Leibniz algebras under certain conditions on their extensions. Additionally, we explore the properties of derivations and direct sums of Leibniz algebras, proving several results analogous to those in Lie algebras. [ABSTRACT FROM AUTHOR]

We show that all finite Brouwerian semilattices have strong endomorphism kernel property (SEKP), give a new proof that all finite relative Stone algebras have SEKP and also fully characterize dual generalized Boolean algebras which possess SEKP.

As a proper generalization of comultiplication modules, the concept of k-comultipli-cation semimodules is introduced in this paper. Let S be a semiring. An left S-semimodule M is called a left k-comultiplication semimodule if for each subtractive subsemimodule N of M, there exists an ideal I of S such that N = (0 : M r I) . Some properties of k-comultiplication semimodules are obtained and some conditions are given for a direct sum of k-comultiplication semimodules to be a k-comultiplication semimodule. [ABSTRACT FROM AUTHOR]

Locally repairable codes (LRCs), which can recover any symbol of a codeword by reading only a small number of other symbols, have been widely used in real-world distributed storage systems, such as Microsoft Azure Storage and Ceph Storage Cluster. Since binary linear LRCs can significantly reduce coding and decoding complexity, constructions of binary LRCs are of particular interest. The aim of this paper is to construct dimensional optimal binary LRCs with disjoint local repair groups. We introduce a method to connect intersection subspaces with binary LRCs and construct dimensional optimal binary linear LRCs with locality 2b (b ≽ 3) and minimum distance d ≽ 6 by employing intersection subspaces deduced from the direct sum. This method will sufficiently increase the number of possible repair groups of dimensional optimal LRCs, thus efficiently expanding the range of the construction parameters while keeping the largest code rates compared with all known binary linear LRCs with minimum distance d ≽ 6 and locality 2b. [ABSTRACT FROM AUTHOR]

In this paper, we provide the basic properties of (semi)simple hypermodules. We show that if a hypermodule M is simple, then (E n d (M) , ·) is a group, where E n d (M) is the set of all normal endomorphisms of M. We prove that every simple hypermodule is normal projective with a zero singular subhypermodule. We also show that the class of semisimple hypermodules is closed under internal direct sums, factor hypermodules, and subhypermodules. In particular, we give a characterization of internal direct sums of subhypermodules of a hypermodule. [ABSTRACT FROM AUTHOR]

It was proved by S. Mazur and S. Ulam in 1932 that every isometric surjection between normed real vector spaces is affine. We generalize the Mazur–Ulam theorem and find necessary and sufficient conditions under which distance-preserving mappings between seminormed real vector spaces are linear. [ABSTRACT FROM AUTHOR]

For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for q-matroids, the q-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of submodular functions and the q-analogue of matroid union we come to a definition of the direct sum of q-matroids. As a motivation for this definition, we show it has some desirable properties. [ABSTRACT FROM AUTHOR]

We show that all finite Brouwerian semilattices have strong endomorphism kernel property (SEKP), give a new proof that all finite relative Stone algebras have SEKP and also fully characterize dual generalized Boolean algebras which possess SEKP. [ABSTRACT FROM AUTHOR]

The concept of t-complement submodule is proposed, which is equivalent to t-closed submodule. The injectivity of direct sum factors of t-extending modules is discussed, the direct sum of t-extending modules is studied, and it is proved that M=⊕i∈IMi(| There are two equivalent conditions for I|≥2) to be a t-extending module: (1) There exists i≠j∈I, so that for any t-closed submodule K of M, if K∩Mi≤Z2(M) or K∩Mj≤Z2(M), then K is the direct sum factor of M ;(2) There exists i≠j∈I. Such that any t-complement of Mj or Mi in M ​​is a t-extending module and is a direct sum of M factor. [ABSTRACT FROM AUTHOR]

In this paper, we study the concept of interval-valued fuzzy set on the family SS X , E of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy topology (cotopology) τ on SS X , E . We obtain that each interval-valued fuzzy topology is a descending family of soft topologies. In addition, we study some topological structures such as interval-valued fuzzy neighborhood system of a soft point, base and subbase of τ and investigate some relationships among them. Finally, we give some concepts such as direct sum, open mapping and continuous mapping and consider connections between them. A few examples support the presented results. [ABSTRACT FROM AUTHOR]

In the given study, we intended to gain familiarity with the idea of fuzzy Hom–Lie subalgebras (ideals) of Hom–Lie algebras. It primarily seeks to study a few of their properties. This research investigates the relationship between fuzzy Hom–Lie subalgebras (ideals) and Hom–Lie subalgebras (ideals). Additionally, this study constructs new fuzzy Hom–Lie subalgebras based on the direct sum of a finite number of existing ones. Finally, the properties of fuzzy Hom–Lie subalgebras and fuzzy Hom–Lie ideals are examined in the context of the morphisms of Hom–Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, we provide the basic properties of (semi)simple hypermodules. We show that if a hypermodule M is simple, then (End(M),·) is a group, where End(M) is the set of all normal endomorphisms of M. We prove that every simple hypermodule is normal projective with a zero singular subhypermodule. We also show that the class of semisimple hypermodules is closed under internal direct sums, factor hypermodules, and subhypermodules. In particular, we give a characterization of internal direct sums of subhypermodules of a hypermodule.

A mixed lattice vector space is a partially ordered vector space with two partial orderings and certain lattice-type properties. In this paper we first give some fundamental results in mixed lattice groups, and then we investigate the structure theory of mixed lattice vector spaces, which can be viewed as a generalization of the theory of Riesz spaces. More specifically, we study the properties of ideals and bands in mixed lattice spaces, and the related idea of representing a mixed lattice space as a direct sum of disjoint bands. Under certain conditions, these decompositions can also be given in terms of order projections. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define the concept of Q-soft R-submodules over commutative rings and discuss their relationship with R-submodules. Next, We describe some of the their basic preoperties and discuss the master properties of them. Later, we introduce the concepts of sum, intersection and external direct sum of them and study some types of separation axioms of them. Finally, we investigate them under homomorphisms of R-submodules. [ABSTRACT FROM AUTHOR]

In this paper, we study sufficient conditions to improve the lower bound on the algebraic immunity of a direct sum of Boolean functions. We exhibit three properties on the component functions such that satisfying one of them is sufficient to ensure that the algebraic immunity of their direct sum exceeds the maximum of their algebraic immunities. These properties can be checked while computing the algebraic immunity and they allow to determine better the security provided by functions central in different cryptographic constructions such as stream ciphers, pseudorandom generators, and weak pseudorandom functions. We provide examples for each property and determine the exact algebraic immunity of candidate constructions. [ABSTRACT FROM AUTHOR]

In this paper, we study the concept of interval-valued fuzzy set on the family SSX,E of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy topology (cotopology) τ on SSX,E. We obtain that each interval-valued fuzzy topology is a descending family of soft topologies. In addition, we study some topological structures such as interval-valued fuzzy neighborhood system of a soft point, base and subbase of τ and investigate some relationships among them. Finally, we give some concepts such as direct sum, open mapping and continuous mapping and consider connections between them. A few examples support the presented results.

Let V be a d -dimensional vector space over a finite field F equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose | F | = q 2 if V is hermitian, and | F | = q otherwise. Given integers e , e ′ such that e + e ′ ⩽ d , we estimate the proportion of pairs (U , U ′) , where U is a non-degenerate e -subspace of V and U ′ is a non-degenerate e ′ -subspace of V , such that U ∩ U ′ = 0 and U ⊕ U ′ is non-degenerate (the sum U ⊕ U ′ is direct and usually not perpendicular). The proportion is shown to be positive and at least 1 − c / q > 0 for some constant c. For example, c = 7 / 4 suffices in both the unitary and symplectic cases. The arguments in the orthogonal case are delicate and assume that dim ⁡ (U) and dim ⁡ (U ′) are even, an assumption relevant for an algorithmic application (which we discuss) for recognising finite classical groups. We also describe how recognising a classical groups G relies on a connection between certain pairs (U , U ′) of non-degenerate subspaces and certain pairs (g , g ′) ∈ G 2 of group elements where U = im (g − 1) and U ′ = im (g ′ − 1). [ABSTRACT FROM AUTHOR]

The parafree Leibniz algebras are a special class of Leibniz algebras which have many properties with a free Leibniz algebra. In this note, we introduce the structure of parafree Leibniz algebras. We survey the important results in parafree Leibniz algebras which are analogs of corresponding results in parafree Lie algebras. We first investigate some properties of subalgebras and quotient algebras of parafree Leibniz algebras. Then, we describe the direct sum of parafree Leibniz algebras. We show that the direct sum of two parafree Leibniz algebras is a Leibniz algebra. Furthermore, we prove that the direct sum of two parafree Leibniz algebras is again parafree. [ABSTRACT FROM AUTHOR]

Let n be a natural number. In this paper we characterize all degree n faithful representations of a dihedral group G of order 2m, m ≥ 3, over the field of complex numbers C. The results are important due to their applications in the study of physical sciences. [ABSTRACT FROM AUTHOR]

In the given study, we intended to gain familiarity with the idea of fuzzy Hom–Lie subalgebras (ideals) of Hom–Lie algebras. It primarily seeks to study a few of their properties. This research investigates the relationship between fuzzy Hom–Lie subalgebras (ideals) and Hom–Lie subalgebras (ideals). Additionally, this study constructs new fuzzy Hom–Lie subalgebras based on the direct sum of a finite number of existing ones. Finally, the properties of fuzzy Hom–Lie subalgebras and fuzzy Hom–Lie ideals are examined in the context of the morphisms of Hom–Lie algebras.

Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the well-known FLIP stream cipher. In this paper, we present a systematic method of constructing weightwise almost perfectly balanced Boolean functions on an arbitrary number of variables, which equal the direct sum of several known weightwise perfectly balanced Boolean functions. At the same time, we show two concrete constructions of weightwise almost perfectly balanced Boolean functions, whose k -weight nonlinearities and algebraic immunities are discussed at the end of this paper. [ABSTRACT FROM AUTHOR]

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter k R -spaces and locally convex spaces E such that E × E is a k R -space. [ABSTRACT FROM AUTHOR]

Minimal codes are characterized by the property that none of the codewords is covered by some other linearly independent codeword. We first show that the use of a bent function g in the so-called direct sum of Boolean functions h (x , y) = f (x) + g (y) , where f is arbitrary, induces minimal codes. This approach gives an infinite class of minimal codes of length 2 n and dimension n + 1 (assuming that h : F 2 n → F 2 ), whose weight distribution is exactly specified for certain choices of f. To increase the dimension of these codes with respect to their length, we introduce the concept of non-covering permutations (referring to the property of minimality) used to construct a bent function g in s variables, which allows us to employ a suitable subspace of derivatives of g and generate minimal codes of dimension s + s / 2 + 1 instead. Their exact weight distribution is also determined. In the second part of this article, we first provide an efficient method (with easily satisfied initial conditions) of generating minimal [ 2 n , n + 1 ] linear codes that cross the so-called Ashikhmin–Barg bound. This method is further extended for the purpose of generating minimal codes of larger dimension n + s / 2 + 2 , through the use of suitable derivatives along with the employment of non-covering permutations. To the best of our knowledge, the latter method is the most general framework for designing binary minimal linear codes that violate the Ashikhmin–Barg bound. More precisely, for a suitable choice of derivatives of h (x , y) = f (x) + g (y) , where g is a bent function and f satisfies certain minimality requirements, for any fixed f, one can derive a huge class of non-equivalent wide binary linear codes of the same length by varying the permutation ϕ when specifying the bent function g (y 1 , y 2) = ϕ (y 2) · y 1 in the Maiorana–McFarland class. The weight distribution is given explicitly for any (suitable) f when ϕ is an almost bent permutation. [ABSTRACT FROM AUTHOR]

Let (A , ‖ · ‖ A) and (B , ‖ · ‖ B) be commutative and semisimple Banach algebras. In this paper, we verify the BED property for the direct sum of A and B , denoted by A ⊕ B . In particular, we show that A ⊕ B is a BED algebra if and only if A and B are so. [ABSTRACT FROM AUTHOR]

The number of inversion sequences avoiding two patterns 101 and 102 is known to be the same as the number of permutations avoiding three patterns 2341, 2431, and 3241. This sequence also counts the number of Schröder paths without triple descents, restricted bicolored Dyck paths, (101 , 021) -avoiding inversion sequences, and weighted ordered trees. We provide bijections to integrate them together by introducing F -paths. Moreover, we define three kinds of statistics for each of the objects and count the number of each object with respect to these statistics. We also discuss direct sums of each object. [ABSTRACT FROM AUTHOR]

Supermixing and hypermixing strongly continuous semigroups are introduced in this paper. It is proved that supermixing preserves under quasiconjugacy. Moreover, it is established that if a strongly continuous semigroup is supermixing(hypermixing), then any discretization of it, is supermixing(hypermixing). Also, it is proved that supermixing(hypermixing) of (Tt)t ≥ 0 implies that any of Tt's is a supermixing(hypermixing). Some sufficient conditions for supermixing and hypermixing are stated that based on dense sets and kernels of operators of semigroup. It is proved that the supermixing(hypermixing) of two strongly continuous semigroups imply that their direct sum is supermixing(hypermixing). Also, supermixing(hypermixing) of the direct sum of finite strongly continuous semigroups indicates that any of these semigroups is supermixing(hypermixing). Outcomes of this study have shown that extension of the concepts of supermixing and hypermixing to strongly continuous semigroups leads to get access to a proper subset of the set of hypercyclic semigroups and reach valuable results about them.

It is well known that super derivation of a Lie superalgebra is certain generalization of derivation of a Lie algebra. This paper is devoted to investigate the structure and dimension of super derivation algebra Der(G) of G where G is a direct sum of two finite dimensional Lie superalgebras L and K having no non-trivial common direct factor. We also introduce some of its sub super algebras. Moreover, we create a condition which shows the isomorphism between super derivation of direct sum and direct sum of super derivations of two Lie superalgebras. Later on, we take G as a semidirect sum of two Lie superalgebras and obtain the structure of Der(G : K) which is a sub super algebra of Der(G) that contains those super derivations mapping K to itself. Finally, we give some conditions under which Der(G : K) is also a semidirect sum. [ABSTRACT FROM AUTHOR]

Categorical constructions on heaps and modules over trusses are considered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coproducts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sum of the group retracts of both heaps and ℤ . Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free modules are constructed as direct sums of a truss with itself. It is shown that only free rank-one module over a ring are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. [ABSTRACT FROM AUTHOR]

We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a finite sum of components, where each component belongs to its own prescribed Banach space; moreover, the problem is regularized by penalizing some composite norm of the solution. We establish general conditions for existence and derive the generic parametric representation of the solution components. These representations fall into three categories depending on the underlying regularization norm: (i) a linear expansion in terms of predefined "kernels" when the component space is a reproducing kernel Hilbert space (RKHS), (ii) a non-linear (duality) mapping of a linear combination of measurement functionals when the component Banach space is strictly convex, and, (iii) an adaptive expansion in terms of a small number of atoms within a larger dictionary when the component Banach space is not strictly convex. Our approach generalizes and unifies a number of multi-kernel (RKHS) and sparse-dictionary learning techniques for compressed sensing available in the literature. It also yields the natural extension of the classical spline-fitting techniques in (semi-)RKHS to the abstract Banach-space setting. [ABSTRACT FROM AUTHOR]

Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem 2.8}). We shall determine the structure of special elements (which are introduced after Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of three lattices, a lattice with exactly one strong element, a lattice which is a direct sum of 2 element lattices with distinguished elements 1 and a lattice which is a direct sum of 2 element lattices with distinguished elements 0, and the sublattice of strong elements is isomorphic to a product of last two mentioned lattices.

A significant area of module theory is the concept of free modules, projective modules and injective modules. The goal of this study is to characterize the projective G-modules under a single-valued neutrosophic set. So we define neutrosophic G-submodule as a generic version of projective G-submodule. It also describes and derives fundamental algebraic properties including quotient space and direct sum of neutrosophic projective G-submodules

In this work we demonstrate the robustness of a real-space approach for the treatment of infinite systems described with periodic boundary conditions that we have recently proposed (Tavernier et al in J Phys Chem Lett 17:7090, 2000). In our approach we extract a fragment, i.e., a supercell, out of the infinite system, and then modifying its topology into the that of a Clifford torus which is a flat, finite and border-less manifold. We then renormalize the distance between two points by defining it as the Euclidean distance in the embedding space of the Clifford torus. With our method we have been able to calculate the reference results available in the literature with a remarkable accuracy, and at a very low computational effort. In this work we show that our approach is robust with respect to the shape of the supercell. In particular, we show that the Madelung constants converge to the same values but that the convergence properties are different. Our approach scales linearly with the number of atoms. The calculation of Madelung constants only takes a few seconds on a laptop computer for a relative precision of about 10 - 6 . [ABSTRACT FROM AUTHOR]

Infinite orthogonal families of regular subspaces of a Kreĭn space exhibit a wide range of behavior. An elementary method is used to show that conditions on sums of projections produce behavior similar to that of orthogonal closed subspaces of a Hilbert space. [ABSTRACT FROM AUTHOR]

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter kR-spaces and locally convex spaces E such that E×E is a kR-space.

Given a set A ⊆ S n of m permutations of { 1 , 2 , ... , n } and a distance function d , the median problem consists in finding the set M (A) of all the permutations that are the "closest" of this set A. In this article we study the automedian case of the problem, i.e. when A = M (A) , under the Kendall- τ distance. We show that automedian sets of permutations are closed under the direct sum operation and also, when some balancing properties are imposed on these sets, under the shuffle operation. These results allow us to derive a parallel algorithm that computes the medians of any separable set of permutations in O (k ! + m n) , where k is the length of its longest inseparable component. [ABSTRACT FROM AUTHOR]

Supermixing and hypermixing strongly continuous semigroups are introduced in this paper. It is proved that supermixing preserves under quasiconjugacy. Moreover, it is established that if a strongly continuous semigroup is supermixing(hypermixing), then any discretization of it, is supermixing(hypermixing). Also, it is proved that supermixing(hypermixing) of (Tt)t ≥ 0 implies that any of Tt's is a supermixing(hypermixing). Some sufficient conditions for supermixing and hypermixing are stated that based on dense sets and kernels of operators of semigroup. It is proved that the supermixing(hypermixing) of two strongly continuous semigroups imply that their direct sum is supermixing(hypermixing). Also, supermixing(hypermixing) of the direct sum of finite strongly continuous semigroups indicates that any of these semigroups is supermixing(hypermixing). Outcomes of this study have shown that extension of the concepts of supermixing and hypermixing to strongly continuous semigroups leads to get access to a proper subset of the set of hypercyclic semigroups and reach valuable results about them. [ABSTRACT FROM AUTHOR]

Let A ∈ R n × n be a nonnegative irreducible square matrix and let r (A) be its spectral radius and Perron-Frobenius eigenvalue. Levinger asserted and several have proven that r (t) : = r ((1 − t) A + t A ⊤) increases over t ∈ [ 0 , 1 / 2 ] and decreases over t ∈ [ 1 / 2 , 1 ]. It has further been stated that r (t) is concave over t ∈ (0 , 1). Here we show that the latter claim is false in general through a number of counterexamples, but prove it is true for A ∈ R 2 × 2 , weighted shift matrices (but not cyclic weighted shift matrices), tridiagonal Toeplitz matrices, and the 3-parameter Toeplitz matrices from Fiedler, but not Toeplitz matrices in general. A general characterization of the range of t , or the class of matrices, for which the spectral radius is concave in Levinger's homotopy remains an open problem. [ABSTRACT FROM AUTHOR]

Recently, M. Baker and N. Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields. [ABSTRACT FROM AUTHOR]

When formulating universal properties for objects in a dagger category, one usually expects a universal property to characterize the universal object up to unique unitary isomorphism. We observe that this is automatically the case in the important special case of C ∗ -categories, provided that one uses enrichment in Banach spaces. We then formulate such a universal property for infinite direct sums in C ∗ -categories, and prove the equivalence with the existing definition due to Ghez, Lima and Roberts in the case of W ∗ -categories. These infinite direct sums specialize to the usual ones in the category of Hilbert spaces, and more generally in any W ∗ -category of normal representations of a W ∗ -algebra. Finding a universal property for the more general case of direct integrals remains an open problem. [ABSTRACT FROM AUTHOR]

A significant area of module theory is the concept of free modules, projective modules and injective modules. The goal of this study is to characterize the projective G-modules under a single-valued neutrosophic set. So we define neutrosophic G-submodule as a generic version of projective G-submodule. It also describes and derives fundamental algebraic properties including quotient space and direct sum of neutrosophic projective G-submodules [ABSTRACT FROM AUTHOR]

Redundantly constrained mechanisms have – in general – non-uniquely calculated reactions when modeled as rigid multibody systems (MBSs). However, some of the reactions may be unique. An analogous problem of indeterminacy is also present in overactuated MBSs. This paper discusses the constraint-matrix-based method for the uniqueness analysis of the reactions and driving forces (torques) for MBSs with nonholonomic constraints. Four approaches are studied: The rank comparison, SVD, QR, and nullspace methods. The uniqueness criteria are written in a new way. The equivalence of the SVD, QR, and nullspace methods is shown. It is also presented how to check the uniqueness of the selected elements (reactions, driving forces, or more complex combinations) and their individual components. Subsequently, the impact of the driving constraints on the uniqueness of the joint reactions is discussed. Next, the uniqueness analysis using these three methods is extended to perform a newly proposed body-wise analysis instead of the usual constraint-wise analysis. Two examples of spatial systems (one with nonholonomic constraints) are considered to illustrate the approach. Moreover, the computational efficiency of selected methods is analyzed. • A unified formulation of the existing constraint-matrix-based methods is proposed. • Methods for checking the uniqueness of individual force components are presented. • The influence of driving constraints on the uniqueness of reactions is discussed. • The equivalence of constraint analysis methods (SVD, QR, and nullspace) is proven. • A new, body-wise instead of joint-wise, paradigm of uniqueness analysis is devised. [ABSTRACT FROM AUTHOR]