identification of a single factor model using graphical gaussian rules we state a sufficient condition for the global identification of a single factor model when some conditional associations among residuals are allowed the condition relies on the structure of the conditional independence graph of the observed variable given the latent factor and can be derived using graphical rules

reflection group presentations arising from cluster algebras we give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system

automorphisms of graph groups abstract given a graph v e the graph group on is the group generated by the vertex set v with the defining relations being that two adjacent vertices commute in the following the centralizer problem for graph groups is solved and it is shown that for some graph groups aut f is finitely generated by generators which are natural analogues to the nielsen automorphisms for a free group

structure of basic lie superalgebras and of their affine extensions we generalize to the case of superalgebras several properties of simple lie algebras involving the use of dynkin diagrams if to a simple lie algebra can be associated one dynkin diagram it is a finite set of nonequivalent ones which can be constructed for a basic superalgebra or b s a the knowledge of these diagrams which can be obtained for each b s a in a systematic way allows us to deduce the regular subsuperalgebras of a b s a the symmetries of the dynkin diagrams are related to outer automorphisms of b s a and lead to some singular subsuperalgebras finally we consider the extended dynkin diagrams in order to classify the affine b s a and use their symmetries to construct the twisted basic superalgebras

involutary expressions for elements in gln z and sln z abstract we shall give upper bounds for the sets of minimal numbers of involutary factors of elements in the general linear group gl n z and the special linear group sl n z over the ring of rational integers z 0 1 2 for n 2 respectively

bayesian inference for graphical factor analysis models we generalize factor analysis models by allowing the concentration matrix of the residuals to have nonzero off diagonal elements the resulting model is named graphical factor analysis model allowing a structure of associations gives information about the correlation left unexplained by the unobserved variables which can be used both in the confirmatory and exploratory context we first present a sufficient condition for global identifiability of this class of models with a generic number of factors thereby extending the results in stanghellini 1997 and vicard 2000 we then consider the issue of model comparison and show that fast local computations are possible for this purpose if the conditional independence graphs on the residuals are restricted to be decomposable and a bayesian approach is adopted to achieve this aim we propose a new reversible jump mcmc method to approximate the posterior probabilities of the considered models we then study the evolution of political democracy in 75 developing countries based on eight measures of democracy in two different years

hecke algebras of finite type are cellular let mathcal h be the one parameter hecke algebra associated to a finite weyl group w defined over a ground ring in which bad primes for w are invertible using deep properties of the kazhdan lusztig basis of mathcal h and lusztig s a function we show that mathcal h has a natural cellular structure in the sense of graham and lehrer thus we obtain a general theory of specht modules for hecke algebras of finite type previously a general cellular structure was only known to exist in types an and bn

equational theories of semigroups with involution abstract we employ the techniques developed in an earlier paper to show that involutory semigroups arising in various contexts do not have a finite basis for their identities among these are partition semigroups endowed with their natural inverse involution including the full partition semigroup c n for n 2 the brauer semigroup b n for n 4 and the annular semigroup a n for n 4 n even or a prime power also all of these semigroups as well as the jones semigroup j n for n 4 turn out to be inherently nonfinitely based when equipped with another involution the skew one finally we show that similar techniques apply to the finite basis problem for existence varieties of locally inverse semigroups

on the identification of a single factor model with correlated residuals summary a necessary and sufficient condition for the identification of a single factor model with correlated residuals is derived by studying the zero elements of their concentration matrix in particular the new condition is expressed in terms of the complementary graph of the residuals this graphical condition can be simply checked by means of an efficient algorithm an example is also given showing that sometimes the single factor model with correlated residuals can be used to overcome the problem of non identification of a larger factor model

graphical identifiability criteria for causal effects in studies with an unobserved treatment response variable summary we consider the problem of using data in studies with an unobserved treatment response variable in order to evaluate average causal effects when cause effect relationships between variables can be described by a directed acyclic graph and the corresponding recursive factorization of a joint distribution the paper proposes graphical criteria to test whether average causal effects are identifiable even if a treatment response variable is unobserved if the answer is affirmative we provide further formulations for average causal effects from the observed data the graphical criteria enable us to evaluate average causal effects when it is difficult to observe a treatment response variable

cellularity of diagram algebras as twisted semigroup algebras abstract the temperley lieb and brauer algebras and their cyclotomic analogues as well as the partition algebra are all examples of twisted semigroup algebras we prove a general theorem about the cellularity of twisted semigroup algebras of regular semigroups which allows us to easily reproduce the cellularity of these algebras this theorem generalizes a result of east about the cellularity of semigroup algebras of inverse semigroups

on groups generated by involutions of a semigroup abstract an involution on a semigroup s or any algebra with an underlying associative binary operation is a function s s that satisfies x y y x and x x for all x y s the set i s of all such involutions on s generates a subgroup c s i s of the symmetric group sym s on the set s we investigate the groups c s for certain classes of semigroups s and also consider the question of which groups are isomorphic to c s for a suitable semigroup s

idempotent rank in finite full transformation semigroups the subsemigroup sing n of singular elements of the full transformation semigroup on a finite set is generated by n n l 2 idempotents of defect one in this paper we extend this result to the subsemigroup k n r consisting of all elements of rank r or less we prove that the idempotent rank defined as the cardinality of a minimal generating set of idempotents of k n r is s n r the stirling number of the second kind

maximal subgroups of free idempotent generated semigroups over the full transformation monoid let tn be the full transformation semigroup of all mappings from the set 1 n to itself under composition let e e tn denote the set of idempotents of tn and let e e be an arbitrary idempotent satisfying im e r n 2 we prove that the maximal subgroup of the free idempotent generated semigroup over e containing e is isomorphic to the symmetric group sr 2000 mathematics subject classification 20m05 05e15 20f05

partial symmetry reflection monoids and coxeter groups abstract this is the first of a series of papers in which we initiate and develop the theory of reflection monoids motivated by the theory of reflection groups the main results identify a number of important inverse semigroups as reflection monoids introduce new examples and determine their orders

idempotents and one sided units lattice invariants and a semigroup of functors on the category of monoids abstract for a monoid m we denote by g m the group of units e m the submonoid generated by the idempotents and g l m and g r m the submonoids consisting of all left or right units writing m for the monoidal category of monoids g e g l and g r are all monoidal functors m m there are other natural functors associated to submonoids generated by combinations of idempotents and one or two sided units the above functors generate a monoid with composition as its operation we show that this monoid has size 15 and describe its algebraic structure we also show how to associate certain lattice invariants to a monoid and classify the lattices that arise in this fashion a number of examples are discussed throughout some of which are essential for the proofs of the main theoretical results

partial mirror symmetry lattice presentations and algebraic monoids this is the second in a series of papers that develops the theory of reflection monoids motivated by the theory of reflection groups reflection monoids were first introduced in everitt and fountain adv math 223 2010 1782 1814 in this paper we study their presentations as abstract monoids along the way we also find general presentations for certain join semilattices as monoids under join which we interpret for two special classes of examples the face lattices of convex polytopes and the geometric lattices particularly the intersection lattices of hyperplane arrangements another spin off is a general presentation for the renner monoid of an algebraic monoid which we illustrate in the special case of the classical algebraic monoids

on the identification of path analysis models with one hidden variable summary we study criteria for identifiability of path analysis models with one hidden variable we first derive sufficient criteria for identification of models in which marginalisation is carried out over the hidden variable the sufficient criteria are based on the structure of the directed acyclic graph associated with the path analysis model and can be derived from the graph we treat further the identification of models when the hidden variable is conditioned on and establish connections with the extended skew normal distribution finally it is shown that the derived conditions extend the existing graphical criteria for identification

biordered sets come from semigroups information about a semigroup can often be gleaned from its partial algebra of idempotents for example the idempotents of an inverse semigroup form a semilattice all isomorphisms between principal ideals of a semilattice e form the munn inverse semigroup t which contains the fundamental image of every inverse semigroup whose idempotents form the semilattice e ll 123 thus semilattices give rise to all fundamental inverse semigroups successful efforts have been made to generalize the munn construction to the wider class of regular semigroups 1 6 9 13 141 nambooripad achieved this using the concept of a regular biordered set the biordered set of a semigroup s means simply the partial algebra consisting of the set e e s of idempotents of s with multiplication restricted to

the semigroup generated by the idempotents of a partition monoid we study the idempotent generated subsemigroup of the partition monoid in the finite case this subsemigroup consists of the identity and all the singular partitions in the infinite case the subsemigroup is described in terms of certain parameters that measure how far a partition is from being a permutation as one of several corollaries we deduce howie s description from 1966 of the semigroup generated by the idempotents of a full transformation semigroup

representations of the q rook monoid the q rook monoid in q is a semisimple algebra over c q that specializes when q 1 to c rn where rn is the monoid of n n matrices with entries from 0 1 and at most one nonzero entry in each row and column when q is specialized to a prime power in q is isomorphic to the iwahori algebra hc m b where m mn fq is the monoid of n n matrices with entries from a finite field having q elements and b m is the borel subgroup of invertible upper triangular matrices in this paper we i give a new presentation for in q on generators and relations and determine a set of standard words which form a basis ii explicitly construct a complete set of seminormal irreducible representations of in q and iii show that in q is the centralizer of the quantum general linear group uqgl r acting on the tensor product w v n where v is the fundamental uqgl r module and w is the trivial uqgl r module

quiver mutations and boolean reflection monoids abstract in 2010 everitt and fountain introduced the concept of reflection monoids the boolean reflection monoids form a family of reflection monoids symmetric inverse semigroups are boolean reflection monoids of type a in this paper we give a family of presentations of boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of dynkin diagrams with frozen vertices our results recover the presentations of boolean reflection monoids given by everitt and fountain and the presentations of symmetric inverse semigroups given by popova respectively surprisingly inner by diagram automorphisms of irreducible weyl groups and boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams besides we show that semigroup algebras of boolean reflection monoids are cellular algebras

generators and relations for partition monoids and algebras abstract we investigate the manner in which the partition monoid p n and algebra p n may be presented by generators and relations making use of structural properties of p n as well as presentations for several key submonoids we obtain a number of presentations for p n including that given without a complete proof by halverson and ram in 2005 we then conclude by showing how each of these presentations gives rise to an algebra presentation for p n

on identification of multi factor models with correlated residuals summary we specify some conditions for the identification of a multi factor model with correlated residuals uncorrelated factors and zero restrictions in the factor loadings these conditions are derived from the results of stanghellini 1997 and vicard 2000 which deal with single factor models with zero restrictions in the concentration matrix like these authors we make use of the complementary graph of residuals and the conditions build on the role of odd cycles in this graph however in contrast to these authors we consider the case where the conditional dependencies of the residuals are expressed in terms of a covariance matrix rather than its inverse the concentration matrix we first derive the corresponding condition for identification of single factor models with structural zeros in the covariance matrix of the residuals this is extended to the case where some factor loadings are constrained to be zero we use these conditions to obtain a sufficient and a necessary condition for identification of multi factor models

realization of a class of affine lie superalgebras in this paper a class of affine lie superalgebras is presented whose dynkin diagrams are constructed by as usual extending a dynkin diagram d of lie algebra or superalgebra by adding a node that corresponds to the lowest root 0 of d however this added node does not have the same degree as 0 in z2 gradation z2 z 2z i e corresponds to an odd or even root as 0 corresponds to even or odd root as far as we know these lie superalgebras did not appear in literature and therefore we believe these are a new class of affine lie superalgebras

representation of the affine superalgebras a 4 0 2l a 2 0 2l 1 and their subalgebras a2l 2 a2l 1 2 by vertex operators abstract the structure theory of standard modules of affine lie algebras given by j lepowsky and r l wilson in lw is stated for representations of affine superalgebras as an application the standard modules of level one for the superalgebras a 4 0 2l a 2 0 2l 1 and their affine subalgebras a2l 2 a2l 1 2 are constructed explicitly these modules are realized as the tensor product of symmetric and exterior algebras with an irreducible representation of a certain finite 2 group the affine superalgebra acts on this space by tensor products of vertex operators operators of clifford type and elements of the 2 group as a corollary the spin representations of the lie algebras bl and dl are obtained from the 2 group representation

on recovering a population covariance matrix in the presence of selection bias this paper considers the problem of using observational data in the presence of selection bias to identify causal effects in the framework of linear structural equation models we propose a criterion for testing whether or not observed statistical dependencies among variables are generated by conditioning on a common response variable when the answer is affirmative we further provide formulations for recovering the covariance matrix of the whole population from that of the selected population the results of this paper provide guidance for reliable causal inference based on the recovered covariance matrix obtained from the statistical information with selection bias

motzkin monoids and partial brauer monoids we study the partial brauer monoid and its planar submonoid the motzkin monoid we conduct a thorough investigation of the structure of both monoids providing information on normal forms green s relations regularity ideals idempotent generation minimal idempotent generating sets and so on we obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent generated we find formulae for the rank smallest size of a generating set of each ideal and for the idempotent rank smallest size of an idempotent generating set of the idempotent generated subsemigroup of each ideal in particular when an ideal is idempotent generated the rank and idempotent rank are equal along the way we obtain a number of results of independent interest and we demonstrate the utility of the semigroup theoretic approach by applying our results to obtain new proofs of some important representation theoretic results concerning the corresponding diagram algebras the partial or rook brauer algebra and motzkin algebra

idempotents and one sided units in infinite partial brauer monoids abstract we study monoids generated by various combinations of idempotents and one or two sided units of an infinite partial brauer monoid this yields a total of eight such monoids each with a natural characterisation in terms of relationships between parameters associated to brauer graphs we calculate the relative ranks of each monoid modulo any other such monoid it may contain and then apply these results to determine the sierpinski rank of each monoid and ascertain which ones have the semigroup bergman property we also make some fundamental observations about idempotents and units in arbitrary monoids and prove some general results about relative ranks for submonoids generated by these sets

braids and partial permutations abstract we study the inverse braid monoid ib n introduced by easdown and lavers in 2004 we completely describe the factorizable structure of ib n and use this to give a new proof of the easdown lavers presentation we also derive several new presentations each of which gives rise to a new presentation of the symmetric inverse monoid we then define and study the pure inverse braid monoid ip n which is related to ib n in the same way that the pure braid group is related to the braid group

on regularity and the word problem for free idempotent generated semigroups the category of all idempotent generated semigroups with a prescribed structure e of their idempotents e called the biordered set has an initial object called the free idempotent generated semigroup over e defined by a presentation over alphabet e and denoted by ig e recently much effort has been put into investigating the structure of semigroups of the form ig e especially regarding their maximal subgroups in this paper we take these investigations in a new direction by considering the word problem for ig e we prove two principal results one positive and one negative we show that for a finite biordered set e it is decidable whether a given word w e represents a regular element if in addition one assumes that all maximal subgroups of ig e have decidable word problems then the word problem in ig e restricted to regular words is decidable on the other hand we exhibit a biorder e arising from a finite idempotent semigroup s such that the word problem for ig e is undecidable even though all the maximal subgroups have decidable word problems this is achieved by relating the word problem of ig e to the subgroup membership problem in finitely presented groups