principal minors and diagonal similarity of matrices abstract let a b be n n matrices with entries in a field f our purpose is to show the following theorem suppose n 4 a is irreducible and for every partition of 1 2 n into subsets with 2 2 either rank a 2 or rank a 2 if a and b have equal corresponding principal minors of all orders then b or b t is diagonally similar to a

the principal rank characteristic sequence of a real symmetric matrix abstract given a vector u r 2 n the principal minor assignment problem asks when is there an n n matrix having its 2 n principal minors given by u this paper explores the following related problem given a sequence r 0 r 1 r n of 0s and 1s does there exist an n n real symmetric matrix that has a principal submatrix of rank k if and only if r k 1 for all 0 k n certain conditions are shown to be necessary in order for this question to have an affirmative answer several families of matrices are constructed to attain certain classes of sequences the problem is solved completely for n 6 and for 7 n 10 in the case of sequences beginning with 010

fuzzy filters of bl algebras the aim of this paper is to introduce the notions of fuzzy filters fuzzy prime filters and the cosets of a fuzzy filter in bl algebras and investigate some of their properties the fuzzy filter generated by a fuzzy set is discussed some characterizations of fuzzy filters and fuzzy prime filters are derived the extension theorem of fuzzy prime filters and the fuzzy prime filters theorem are established finally we prove that the algebra l f which is the set of all cosets of f is a bl algebra and is isomorphic to the bl algebra l ff 1 where ff 1 x l f x f 1 moreover each bl algebra l is a subdirect product of linearly ordered bl algebras l fi i where fi i are fuzzy prime filters of l

polynomial relations among principal minors of a 4x4 matrix abstract the image of the principal minor map for n n matrices is shown to be closed in the 19th century nanson and muir studied the implicitization problem of finding all relations among principal minors when n 4 we complete their partial results by constructing explicit polynomials of degree 12 that scheme theoretically define this affine variety and also its projective closure in p 15 the latter is the main component in the singular locus of the 2 2 2 2 hyperdeterminant

principal minor assignment isometries of hilbert spaces volumes of parallelepipeds and rescaling of sesqui holomorphic functions abstract in this article we consider the following equivalence relation on the class of all functions of two variables on a set x we will say that l m x x c are rescalings if there are non vanishing functions f g on x such that m x y f x g y l x y for any x y x we give criteria for being rescalings when x is a topological space and l and m are separately continuous or when x is a domain in c n and l and m are sesqui holomorphic a special case of interest is when l and m are symmetric and f g only has values 1 this relation between m and l in the case when x is finite and so l and m are square matrices is known to be characterized by the equality of the principal minors of these matrices we extend this result to the case when x is infinite as an application we characterize restrictions of isometries of hilbert spaces on weakly connected sets as the maps that preserve the volumes of parallelepipeds spanned by finite collections of vectors

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

primitivity is hereditary for 2 structures abstract it is proved that each primitive 2 structure on n elements n 3 contains a primitive substructure on either n 1 or n 2 elements

hyperfocal subalgebras of source algebras of blocks over small ground fields 1 1 the local theory of blocks of finite groups was proposed originally by j alperin and m broue in 1 and developed by l puig 12 where the source algebra of a block is introduced as the smallest algebra which carries the local information of the block one of the classical applications of the theory is the research on nilpotent blocks see 2 9 recently understanding the fusions of local pointed groups l puig in 7 and 8 introduces the hyperfocal subalgebra in the source algebra of a block and proves its existence and uniqueness up to conjugation the local information of nilpotent blocks are the simplest case and the structure theorem of their source algebras in 9 is the simplest case of the puig s work on hyperfocal subalgebras noting that puig obtains his results in large enough coefficient fields in this paper we make a research on the hyperfocal subalgebras of source algebras of blocks over small ground fields

a matrix description of weakly bipartitive and bipartitive families abstract the notions of weakly bipartitive and bipartitive families were introduced by montgolfier 2003 as a general tool for studying some decomposition of graphs and other combinatorial structures one way to construct such families comes from a result of loewy 1986 given an irreducible n n matrix a over a field the family of partitions x y of 1 n such that the submatrices a x y and a y x have a rank at most 1 is weakly bipartitive in this paper we show that this family is bipartitive when a is symmetric in the converse direction we prove that weakly bipartitive and bipartitive families are all obtained via the construction above

fuzzy implicative and boolean ideals of mv algebras abstract we show how to obtain results in mv algebras by considering their fuzzy ideals in particular we prove the following theorem let be a fuzzy ideal of an mv algebra x then the folllowing are equivalent 1 x x 0 for all x x 2 x n x for all x x and all n 1 3 x x is a boolean algebra 4 is fuzzy implicative if i is an ideal of x and we take i then we obtain corresponding results for the ideal i and x we show how to extend fuzzy prime and fuzzy implicative ideals various fuzzy radicals are considered and related to the corresponding radicals of the algebra and we show how they live in the spectrum of the algebra

the socle and fuzzy socle of a bci algebra we introduce the notion of orthogonal complement or annihilator in general bci algebras and the corresponding notion in the fuzzy ideals various properties are proved and the related notions of the socle and fuzzy socle of a bci algebra are investigated it is shown that a non zero weakly commutative artinian bci algebra has non zero socle

an efficient algorithm for the symmetric principal minor assignment problem abstract we consider the problem of reconstructing a symmetric matrix from its principal minors which has several applications in information theory and statistical modeling we develop a theory of symmetric matrices with equal corresponding principal minors based on a simple equivalent property due to oeding 2011 10 we then use this theory to provide a method for choosing a canonical representative from the class of all symmetric matrices with specified principal minors finally we provide an efficient algorithm for computing this canonical representative given its principal minors as input

skew symmetric matrices and their principal minors abstract in this article we study the following problem given a positive integer k what is the relationship between two matrices with entries in a field k and having equal corresponding principal minors of order at most k our main theorem improves a result of loewy 13 for skew symmetric matrices all of whose off diagonal entries are nonzero

hyperdeterminantal relations among symmetric principal minors abstract the principal minors of a symmetric n n matrix form a vector of length 2 n we characterize these vectors in terms of algebraic equations derived from the 2 2 2 hyperdeterminant

on group stable commutative separable semisimple subalgebras in a g algebra a of a p group g over a perfect ground field there is a commutative separable semisimple g subalgebra such that any traces on relative projectivity in a can be controlled in the subalgebra and in a residually central sense the maximal such subalgebras form exactly one a g conjugacy class applying to modules the induced module of an indecomposable module of a subnormal subgroup of p primary index is characterized

principal minors part i a method for computing all the principal minors of a matrix an order o 2 n algorithm for computing all the principal minors of an arbitrary n n complex matrix is motivated and presented offering an improvement by a factor of n 3 over direct computation the algorithm uses recursive schur complementation and submatrix extraction storing the answer in a binary order an implementation of the algorithm in matlab is also given and practical considerations are discussed and treated accordingly

fuzzy maximal ideals of bci and mv algebras abstract we consider fuzzy maximal ideals of bci bck and mv algebras and show that such fuzzy ideals take only the values 0 1 and have level ideals which are maximal ideals of the algebra it is shown that fuzzy maximal ideals of commutative bck algebras are fuzzy prime we establish a correspondence between ideals and fuzzy ideals in a bounded bck algebra this correspondence is one to one between maximal ideals and fuzzy maximal ideals we also show how to construct fuzzy ideals which are not characteristic functions of ideals of an mv algebra

fuzzy ideals of bci and mv algebras abstract using fuzzy ideals it is shown that every ideal of a bci algebra is weakly implicative for a fuzzy ideal of an mv algebra it is shown that it satisfies x v y x y x y for all x y if it is fuzzy implicative and fuzzy prime then it also satisfies x y x y x x 1 and x x 0 for all x and y and in fact im 0 1

principal minors part ii the principal minor assignment problem the inverse problem of finding a matrix with prescribed principal minors is considered a condition that implies a constructive algorithm for solving this problem will always succeed is presented the algorithm is based on reconstructing matrices from their principal submatrices and schur complements in a recursive manner consequences regarding the overdeterminancy of this inverse problem are examined leading to a faster polynomial time version of the algorithmic construction care is given in the matlab implementation of the algorithms regarding numerical stability and accuracy

on matrices having equal corresponding principal minors abstract let a b be n n matrices with entries in a field f we say a and b satisfy property d if b or b t is diagonally similar to a it is clear that if a and b satisfy property d then they have equal corresponding principal minors of all orders the question is to what extent the converse is true there are examples which show the converse is not always true we modify the problem slightly and give conditions on a matrix a which guarantee that if b is any matrix which has the same principal minors as a then a and b will satisfy property d these conditions on a are formulated in terms of ranks of certain submatrices of a and the concept of irreducibility