monoids on groups with bn pairs the theory of linear algebraic monoids has been developed in the last few years by l renner and the author 21 361 as a generalization of the theory of linear algebraic groups 16 40 in applications of the theory of linear algebraic groups one is naturally led to consider groups with bnpairs or tits systems 12 4 9 42 431 this class of groups includes in particular the groups of lie type so that the abstract properties of tits systems are useful in studying the conjugacy classes and representations of groups of lie type see 4 9 our interest is in monoids so let m be a monoid such that the group of units g has a bn pair in general g has very little to do with the rest of m in fact for a given g m g could be an arbitrary semigroup so we introduce three axioms which intimately connect the idempotent structure of m with the structure of g we then say that m is a monoid on g the prime example is a connected regular linear algebraic monoid with zero but there are other natural examples for instance renner 37 is developing a theory of monoids of lie type along the lines of steinberg 41 for groups another example arises from the work of smith 39 and cabanes 3 from their work one can associate with certain irreducible modular representations 4 g gl v a set of idempotents es end v then the monoid m generated by 4 g and e is a monoid on 4 g in the examples we have worked out and very likely in general in the situation of linear algebraic monoids there is a natural type map from the finite lattice of conjugacy classes of idempotents g the lattice of f classes into the power set of the dynkin diagram of g the idea of the type map is due to lex renner and is exploited by renner and the author 31 to show in particular that the system of idempotents can be constructed from the type map in the present situation of monoids on g there is again a type map i which characterizes the system of idempotents 139 0021 8693 89 3 00

maximal subgroups of free idempotent generated semigroups over the full linear monoid we show that the rank r component of the free idempotent generated semigroup of the biordered set of the full linear semigroup full of n times n matrices over a division ring q has maximal subgroup isomorphic to the general linear group gl r q where n and r are positive integers with r n 3

fixed point theory for 1 set contractive and pseudocontractive mappings the purpose of this paper is to study the existence and uniqueness of fixed point for a class of nonlinear mappings defined on a real banach space which among others contains the class of separate contractive mappings as well as to see that an important class of 1 set contractions and of pseudocontractions falls into this type of nonlinear mappings as a particular case we give an iterative method to approach the fixed point of a nonexpansive mapping later on we establish some fixed point results of krasnoselskii type for the sum of two nonlinear mappings where one of them is either a 1 set contraction or a pseudocontraction and the another one is completely continuous which extend or complete previous results in the last section we apply such results to study the existence of solutions to a nonlinear integral equation

hecke algebras and semisimplicity of monoid algebras abstract we begin by constructing hecke algebras for arbitrary finite regular monoids m we then show that the semisimplicity of the complex monoid algebra c m is equivalent to the semisimplicity of the associated hecke algebras and a condition on induced group characters we apply these results to finite lie type monoids m we show that the monoid algebra fm over a field f is semisimple if and only if the characteristic of f does not divide the order of the unit group g this is accomplished by developing formulas for the unities of c j j a j class of m the unity is explicitly given when g is a simply connected chevalley group and j is associated with a borel subgroup of g

some fixed point theorems of the schauder and the krasnosel skii type and application to nonlinear transport equations abstract in j math phys 37 1996 1336 1348 the existence of solutions to the boundary value problem 1 1 1 2 was analyzed for isotropic scattering kernels on l p spaces for p 1 due to the lack of compactness in l 1 spaces the problem remains open for p 1 the purpose of this work is to extend this analysis to the case p 1 for anisotropic scattering kernels our strategy consists in establishing new variants of the schauder and the krasnosel skii fixed point theorems in general banach spaces involving weakly compact operators in l 1 context these theorems provide an adequate tool to attack the problem our analysis uses the specific properties of weakly compacts sets on l 1 spaces and the weak compactness results for one dimensional transport equations established in j math anal appl 252 2000 767 789

integrable solutions of a generalized mixed type functional integral equation in this work we prove the existence of integrable solutions for the following generalized mixed type nonlinear functional integral equation x t g t t x t f t 0 t k t s u t s q x s d s t 0 our result is established by means of a krasnosel skii type fixed point theorem proved by taoudi 2009 in the last section we give an example to illustrate our result

existence of fixed points and measures of weak noncompactness abstract the purpose of this paper is to study the existence of fixed points by using measures of weak noncompactness later on we provide an existence principle for solutions for a nonlinear integral equation

stability of the solutions of a convection diffusion equation abstract a class of non newtonian fluid equations with a convection term is considered if the diffusion coefficient is degenerate on the boundary the interaction between the convection and diffusion is investigated and the stability of solutions is established without any boundary condition

conjugacy classes of renner monoids abstract in this paper we describe conjugacy classes of a renner monoid r with unit group w the weyl group we show that every element in r is conjugate to an element ue where u w and e is an idempotent in a cross section lattice denote by w e and w e the centralizer and stabilizer of e in w respectively let w e act by conjugation on the set of left cosets of w e in w we find that ue and ve u v w are conjugate if and only if u w e and v w e are in the same orbit as consequences there is a one to one correspondence between the conjugacy classes of r and the orbits of this action we then obtain a formula for calculating the number of conjugacy classes of r and describe in detail the conjugacy classes of the renner monoid of some j irreducible monoids we then generalize munn conjugacy on a rook monoid to any renner monoid and show that munn conjugacy coincides with semigroup conjugacy action conjugacy and character conjugacy we also show that the number of inequivalent irreducible representations of r over an algebraically closed field of characteristic zero equals the number of munn conjugacy classes in r

complex representations of matrix semigroups let m be a finite monoid of lie type these are the finite analogues of linear algebraic monoids with group of units g the multiplicative semigroup 4 f where f is a finite field is a particular example using harishchandra s theory of cuspidal representations of finite groups of lie type we show that every complex representation of m is completely reducible using this we characterize the representations of g extending to irreducible representations of m as being those induced from the irreducible representations of certain parabolic subgroups of g we go on to show that if f is any field and s any multiplicative subsemigroup of 4 f then the semigroup algebra of s over any field of characteristic zero has nilpotent jacobson radical if s 4 f then this algebra is jacobson semisimple finally we show that the semigroup algebra of 4 f over a field of characteristic zero is regular if and only if ch f p 0 and f is algebraic over its prime field

sandwich matrices solomon algebras and kazhdan lusztig polynomials sandwich matrices have proved to be of importance in semigroup theory for the last 50 years the work of the author on algebraic monoids leads to sandwich matrices in group theory in this paper we find some connections between sandwich matrices and the hecke algebras for monoids introduced recently by louis solomon at the local level we then obtain an explicit isomorphism between solomon s hecke algebra and the complex monoid algebra of the renner monoid in the simplest case of monoids associated with a borel subgroup we find that the entries of the inverse of the sandwich matrix as well as those of the related structure matrix of solomon s hecke algebra are almost the polynomials r x y associated with the kazhdan lusztig polynomials

the lattice of j classes of j σ irreducible monoids ii abstract a reductive monoid is an algebraic monoid with a reductive unit group generally we have the question of finding the orbits of the unit group of a reductive monoid acting on both sides of the monoid putcha and renner give a recipe to determine the orbits for j irreducible monoids according to the dynkin diagrams we obtain a similar recipe for the question to j irreducible monoids not j irreducible of type d 2 n however there is no similar answer for types a n n 4 and e 2 6 the fixed points of any j irreducible monoid under is a finite reductive monoid we obtain that any such finite reductive monoid is j irreducible then we find the orbits of these monoids under the two sided action of their unit groups

the solutions of a hyperbolic parabolic mixed type equation on half space domain abstract the solutions of the hyperbolic parabolic mixed type equation u t a u div b u in q t r n 0 t are considered where r n r n is half space a new kind of entropy solution to the equation is introduced the paper shows that the convection term div b u determines the explicit boundary value condition if b n 0 0 we can impose the general dirichlet boundary condition u x t 0 x t r n 0 t 0 t which is satisfied in a particular weak sense but if b n 0 0 then no boundary value condition is necessary the solution of the equation is free from any limitation of the boundary value condition

uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle we study the comparison principle for anisotropic degenerate parabolic hyperbolic equations with initial and nonhomogeneous boundary conditions we prove a comparison theorem for any entropy sub and super solution which immediately deduces the l1 contractivity and therefore uniqueness of entropy solutions the method used here is based upon the kinetic formulation and the kinetic techniques developed by lions perthame and tadmor by adapting and modifying those methods to the case of dirichlet boundary problems for degenerate parabolic equations we can establish a comparison property moreover in the quasi isotropic case the existence of entropy solutions is proved

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

the h polynomial of an irreducible representation abstract let g be a simple algebraic group associated with the finite dimensional rational representation g end v of g there is the monoid m k g end v and the projective g g embedding p m 0 k one can identify the cases where p is rationally smooth and in such cases it is desirable to calculate the h polynomial h of p in this paper we consider the situation where is irreducible we then determine h explicitly in terms of combinatorial invariants of indeed there is a canonical cellular decomposition for p these cells are defined in terms of idempotents b b orbits and other natural quantities obtained from m furthermore h is obtained by recording the dimension of each of these cells in terms of the descent system of m as a special case we reacquire the well known formula for the poincare polynomial of a wonderful embedding of a simple algebraic group of adjoint type

the h polynomial of a semisimple monoid abstract a semisimple monoid m is called quasismooth if m 0 has sufficiently mild singularities we define a cellular decomposition of such monoids using the method of one parameter subgroups these cells turn out to be almost affine spaces but they can also be described in terms of the idempotents and b b orbits of m this leads to a number of combinatorial results about the inverse monoid of b b orbits of m in particular we obtain fundamental information about the h polynomial of m

idempotent lattices renner monoids and cross section lattices of the special orthogonal algebraic monoids abstract let mson n is even be the special orthogonal algebraic monoid t a maximal torus of the unit group and t the zariski closure of t in the whole matrix monoid m n in this paper we explicitly determine the idempotent lattice e t the renner monoid r and the cross section lattice of mson in terms of the weyl group and the concept of admissible sets see definition 3 1 it turns out that there is a one to one relationship between e t and the admissible subsets and that r is a submonoid of r n the renner monoid m n also is a sublattice of n the cross section lattice of m n

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

stability of hyperbolic parabolic mixed type equations with partial boundary condition abstract in this paper we are concerned with the hyperbolic parabolic mixed type equations with the non homogeneous boundary condition if it is degenerate on the boundary the part of the boundary whose boundary value should be imposed is determined by the entropy condition from the convection term if there is no convection term in the equation we show that the stability of solutions can be proved without any boundary condition if the equation is completely degenerate we show that the stability of solutions can be established just based on the partial boundary condition

representations of monoids arising from finite groups of lie type a class of finite monoids m constructed from a group g of lie type is considered we describe the irreducible complex representations and prove the complete reducibility of the representations of m the sandwich matrix of m is decomposed into a product of matrices corresponding to maximal parabolic subgroups of g

classification of semisimple rank one monoids consider the classification problem for irreducible normal algebraic monoids with unit group g we obtain complete results for the groups s12 k x k g12 k and pgi2 k x k if g is one of these groups let 4 g denote the set of isomorphy types of normal algebraic monoids with zero element and unit group g our main result establishes a canonical one to one correspondence g q where q is the set of positive rational numbers the classification is achieved in two steps first we construct a class of monoids from linear representations of g that done we show that any other e must already be one of those constructed to do this we devise an extension principle analogous to the big cell construction of algebraic group theory this yields a birational comparison morphism qp er e for some r e q which is ultimately an isomor phism because the monoid er e g is regular

the canonical compactification of a finite group of lie type let g be a finite group of lie type we construct a finite monoid m having g as the group of units m has properties analogous to the canonical compactification of a reductive group the complex representation theory of m yields harish chandra s philosophy of cuspidal representations of g the main purpose of this paper is to determine the irreducible modular representations of m we then show that all the irreducible modular representations of g come via the 1942 work of clifford from the one dimensional representations of the maximal subgroups of m this yields a semigroup approach to the modular representation theory of g via the full rank factorizations of the sandwich matrices of m we then determine the irreducible modular representations of any finite monoid of lie type

partial boundary value condition for a nonlinear degenerate parabolic equation abstract in this paper a nonlinear degenerate parabolic equation is considered by proposing a new partial boundary value condition and constructing special test functions we prove the stability of the solutions by means of the kruzkov bi variables method moreover we show that if there is an interaction between the diffusion term and the convection term the stability can be established without any boundary value condition

conjugacy classes in algebraic monoids let m be a connected linear algebraic monoid with zero and a reductive group of units g the following theorem is established theorem there exist affne subsets ml mk of m reductive groups g1 gk with antiautomorphisms surjective morphisms 0t mi gi such that 1 every element of m is conjugate to an element of some ms and 2 two elements a b in mi are conjugate in m if and only if there exists x e gi such that x0 a x si b as a consequence it is shown that m is a union of its inverse submonoids introduction the objects of study in this paper are connected linear algebraic monoids m with zero this means by definition that the underlying set of m is an irreducible affine variety and that the product map is a morphism i e a polynomial map we will further assume that the group of units g of m is reductive this means 1 3 that the unipotent radical of g is trivial then by 6 10 m is unit regular i e m e m g where e e m e e mle2 e in this paper we study the conjugacy classes of m an initial study was made by the author 8 where the general problem was reduced to nilpotent elements the approach here is quite different yielding a more complete answer to be precise we show that there exist affine subsets m1 mk of m reductive groups g1 gk with antiautomorphisms surjective morphisms si mi gi i 1 k such that 1 every element of m is conjugate to an element of some mi and 2 if a b e mi then a is conjugate to b in m if and only if there exists x e gi such that xsi a x si b as an application of this result we show that m is a union of its inverse submonoids an inverse semigroup is a semigroup s with the property that for each a e s there exists a unique a e s such that aaa a and aaa a see 2 finally in 3 we use our main results to briefly analyze the conjugacy classes of nilpotent elements 1 preliminaries throughout this paper z will denote the set of all positive integers and k an algebraically closed field let g be a connected linear algebraic group defined over k the radical r g is the maximal closed connected normal solvable subgroup of g and the unipotent radical ru g is the group of unipotent elements of r g we will assume throughout that g is a reductive group i e ru g 1 then r g c c g the center of g moreover g r g go where go g g is a semisimple group i e r go 1 also 3 theorem 27 5 go is a product of the simple closed normal subgroups of g in particular we have the following received by the editors november 3 1986 1980 mathematics subject classification 1985 revision primary 20g99 20m10 c 1987 american mathematical society 0002 9947 87 1 00 25 per page

classification of semisimple algebraic monoids let x be a semisimple algebraic monoid with unit group g associated with e is its polyhedral root system x c where x x t is the character group of the maximal torus t c g c x t is the set of roots and c x t is the character monoid of t c e zariski closure the correspondence e x c is a complete and discriminating invariant of the semisimple monoid e extending the well known classification of semisimple groups in establishing this result monoids with preassigned root data are first constructed from linear representations of g that done we then show that any other semisimple monoid must be isomorphic to one of those constructed to do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory thls ultimately yields the desired conclusions consider the classification problem for semisimple algebraic monoids over the algebraically closed field k what sort of problem is this first the definition a semisimple algebraic monoid is an irreducible affine algebraic variety e defined over k together with an associative morphlsm m e x e e and a two sided unit 1 e e for m we assume further that e has a 0 the unit group g which is always linear algebraic and dense in e is reductive e g gl k dim zg 1and e is a normal variety the problem then presents us with two familiar objects let t c g be a maximal torus then z t c e zariski closure is an affine torus embedding and g c e is a reductive algebraic group torus embeddings have been introduced by demazure in 6 in his study of cremona groups and are classified numerically using rational polyhedral cones 14 on the other hand reductive groups have been studied at least in principle since the nineteenth century their classification in the modern sense being achieved largely by chevalley 4 that numerical classification uses the now familiar root systems l l chapter 31 of killing that were introduced by him 15 in his penetrating formulation of the classification e cartan s of semisimple lie algebras thus in the classification of semisimple monoids we are compelled to consider the root system x x t t of g and the polyhedral cone c x z c x t of t c z the two objects are canonically related by the weyl group action on x which leaves c stable received by the editors november 16 1984 1980 mathematics subject classification primary 14m99 secondary 20m99

mobius functions and semigroup representation theory ii character formulas and multiplicities abstract we generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using rota s theory of mobius inversion the technique works for a large class of semigroups including inverse semigroups semigroups with commuting idempotents idempotent semigroups and semigroups with basic algebras using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers these include the random walks on chambers of hyperplane arrangements studied by bidigare hanlon rockmore brown and diaconis applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups generalizing and simplifying earlier results of solomon for the rook monoid

existence results for a nonlinear transport equation in bounded geometry on l1 spaces abstract in the paper latrach 2001 22 the existence of solutions to the boundary value problem 1 2 was discussed on l p spaces for p 1 in bounded geometry due to the lack of compactness on l 1 spaces the analysis did not cover the case p 1 the purpose of this work is to extend the results of 22 to the case p 1 our analysis uses the concept of dunford pettis operators and a new version of the darbo fixed point theorem for the measure of weak noncompactness

homogeneous dirichlet problems for quasilinear anisotropic degenerate parabolic hyperbolic equations the aim of this paper is to prove the well posedness existence and uniqueness of entropy solutions to the general anisotropic degenerate parabolic hyperbolic equations with l initial data and homogeneous dirichlet boundary condition we use the doubling variables device to prove a comparison theorem which implies the uniqueness the existence of entropy solutions can be obtained by finding the limit of solutions for the regularized equation of strictly parabolic type

infinite dimensional reductive monoids associated to highest weight representations of kac moody groups abstract starting with a highest weight representation of a kac moody group over the complex numbers we construct a monoid whose unit group is the image of the kac moody group under the representation multiplied by the nonzero complex numbers we show that this monoid has similar properties to those of a j irreducible reductive linear algebraic monoid in particular the monoid is unit regular and has a bruhat decomposition and the idempotent lattice of the generalized renner monoid of the bruhat decomposition is isomorphic to the face lattice of the convex hull of the weyl group orbit of the highest weight

algebraic rational cells and equivariant intersection theory we provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action based on this notion we study mathbb q filtrable varieties algebraic varieties where a torus acts with isolated fixed points such that the associated bialynicki birula decomposition consists of algebraic rational cells we show that the rational equivariant chow group of any mathbb q filtrable variety is freely generated by the classes of the cell closures we apply this result to group embeddings and more generally to spherical varieties

irreducible representations of semisimple algebraic groups and supersolvable lattices abstract let be the cross section lattice of an irreducible representation of a semisimple algebraic group certain combinatorial properties of are studied supersolvable s are determined in terms of dynkin diagrams the characteristic polynomial of a supersolvable is computed

split subvarieties of group embeddings let g be a connected reductive group and x an equivariant com pactifiction of g in x we study generalised and opposite generalised schu bert varieties their intersections called generalised richardson varieties and projected generalised richardson varieties any complete g embedding has a canonical frobenius splitting and we prove that the compatibly split subvari eties are the generalised projected richardson varieties extending a result of knutson lam and speyer to the situation

representation theory of reductive normal algebraic monoids new results in the representation theory of semisimple algebraic monoids are obtained based on renner s monoid version of chevalley s big cell the semisimple algebraic monoids have been classified by renner the rational representations of such a monoid are the same thing as polynomial representations of the associated reductive group of units in the monoid and this representation category splits into a direct sum of subcategories by homogeneous degree we show that each of these homogeneous subcategories is a highest weight category in the sense of cline parshall and scott and so equivalent with the module category of a certain finite dimensional quasihereditary algebra which we show is a generalized schur algebra in s donkin s sense

existence results for a generalized nonlinear hammerstein equation on l1 spaces abstract in this paper we are concerned with existence results for a nonlinear hammerstein integral equation acting on l 1 vector valued functions making use of the de blasi measure of weak noncompactness we establish a variant of the topological fixed point theorem of krasnosel skii for the sum of two operators without requiring weak compactness or weak continuity assumptions such a result provides an adequate tool for deriving an existence theory for problem 1 1

fixed point methods and accretivity for perturbed nonlinear equations in banach spaces abstract in this paper we use fixed point theorems to guarantee the existence of solutions for inclusions of the form a u u f u g where a is a quasi m accretive operator defined in a banach space 0 and the nonlinear perturbation f satisfies some suitable conditions we apply the obtained results among other things to guarantee the existence of solutions of boundary value problems of the type u x u x f u x g x x and u 0 on where the laplace operator should be understood in the sense of distributions over and to study the existence and uniqueness of solution for a nonlinear integro differential equation posed in l 1

analog of the bruhat decomposition for algebraic monoids ii the length function and the trichotomy 0 introduction the purpose of this paper is to continue the study of b b orbits on a reductive monoidm this was first taken up systematically by the author in 9 where several basic properties analogous to the case of groups were established the b b orbits were identified with a finite monoid r defined in terms of the normalizer of a maximal torus and a direct analysis of these orbits led to an extension of tits axiom br brb b rb to the case of reductive monoids however there is more to this axiom when dealing with monoids obviously

the i class structure of connected algebraic monoids abstract let s be a connected algebraic monoid and let u s denote the finite lattice of all regular i classes of s the main theorem of this paper is that u s is relatively complemented if and only if s is a semilattice of archimedean semigroups if further s has a zero then u s is relatively complemented if and only if the group of units g of s is solvable another characterization for the solvability of g is that the product of any two nilpotent elements of s is again nilpotent the width and depth of any j u s are defined also the width depth and height of s are defined these structure numbers are studied in some detail and the solvability of g is characterized in terms of them

generic hecke algebra for renner monoids abstract to each renner monoid r we associate a generic hecke algebra h r over z q which is a deformation of the monoid z algebra of r if m is a finite reductive monoid with borel subgroup b and associated renner monoid r then we obtain the associated iwahori hecke algebra h m b by specialising q in h r and tensoring by c over z as in the classical case of finite reductive groups

a topological and geometric approach to fixed points results for sum of operators and applications abstract in this paper we establish a fixed point result of krasnoselskii type for the sum a b where a and b are continuous maps acting on locally convex spaces our results extend previous ones we apply such results to obtain strong solutions for some quasi linear elliptic equations with lack of compactness we also provide an application to the existence and regularity theory of solutions to a nonlinear integral equation modeled in a banach space in the last section we develop a sequentially weak continuity result for a class of operators acting on vector valued lebesgue spaces such a result is used together with a geometric condition as the main tool to provide an existence theory for nonlinear integral equations in l p e

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

schaefer krasnoselskii fixed point theorems using a usual measure of weak noncompactness abstract we present some extension of a well known fixed point theorem due to burton and kirk t a burton c kirk a fixed point theorem of krasnoselskii schaefer type math nachr 189 1998 423 431 for the sum of two nonlinear operators one of them compact and the other one a strict contraction the novelty of our results is that the involved operators need not to be weakly continuous finally an example is given to illustrate our results

types of reductive monoids abstract let m be a reductive monoid with a reductive unit group g clearly there is a natural g g action on m the orbits are the j classes in the sense of semigroup theory and form a finite lattice the general problem of finding the lattice remains open in this paper we study a new class of reductive monoids constructed by multilined closure we obtain a general theorem to determine the lattices of these monoids we find that the j irreducible monoids of suzuki type and ree type belong to this new class using the general theorem we then list all the lattices and type maps of the j irreducible monoids of suzuki type and ree type

shellability in reductive monoids the purpose of this paper is to extend to monoids the work of bjorner wachs and proctor on the shellability of the bruhat chevalley order on weyl groups let m be a reductive monoid with unit group g borel subgroup b and weyl group w we study the partially ordered set of b x b orbits with respect to zariski closure inclusion within a g x g orbit of m this is the same as studying a w x w orbit in the renner monoid r such an orbit is the retract of a universal orbit which is shown to be lexicograhically shellable in the sense of bjorner and wachs

irreducible modules and parabolic subgroups summary it seems desirable from the point of view of finite group theorists to develop further the representation theory of finite chevalley groups in the characteristic of their definition say p by focusing attention on the full array of p local subgroups that is the parabolics rather than the usual single borel or cartan subgroup the following observation may be regarded as a generalization to arbitrary parabolic subgroups of the standard result 3 theorem 39 d for a borel subgroup namely that in an irreducible module the subspace fixed by a maximal unipotent subgroup is 1 dimensional and so affords an irreducible module for a levi complement which is an abelian cartan subgroup it seems natural to state the result first for algebraic groups

reductive monoids are von neumann regular let k be an algebraically closed field an a e algebraic monoid e is an affme algebraic variety defined over k together with an associative morphism m e x e e and a two sided unit 1 e e for m in categorical terminology an afftne algebraic monoid is a representable functor from the category of afftne varieties to the category of monoids algebraic group theory is concerned mainly with reductive groups and their representations whereas the theory of monoids is concerned largely with von neumann regular monoids the main result of this paper provides a fundamental link between these two objects

analogue of the bruhat chevalley order for reductive monoids the purpose of this paper is to describe the adherence order of w x b b orbits on a reductive algebraic monoid by the results of 10 there is already a perfect analogue of the bruhat decomposition for reductive w x monoids indeed by 10 corollary 5 8 if m is reductive with unit group g and maximal torus t g with borel subgroup b t then the set of two sided b orbits b r mrb is canonically identified with a certain finite inverse monoid r in fact r s n t rt the orbit monoid of the zariski g closure of n t in m by definition for s t g r we define the adherg ence order by s f t if bs b bt b where bt b denotes the zariski closure in m of bt b

orders of the renner monoids abstract in this paper we give a general formula for a renner monoid of a reductive monoid with zero it is based on putcha s type map we apply this formula to find the orders of the renner monoids for j irreducible monoids k g where g is a simple algebraic group over an algebraically closed field k and g gl v is an irreducible representation associated with an arbitrary fundamental dominant weight