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Assume that A and G are finite groups of coprime orders such that A acts on G via automorphisms. Let p be a prime. The following coprime action version of a well-known theorem of Ito about the structure of a minimal non-p-nilpotent groups is proved: if every maximal A-invariant subgroup of G is p-nilpotent, then G is p-soluble. If, moreover, G is not p-nilpotent, then G must be soluble. Some earlier results about coprime action are consequences of this theorem.

In the preceding article “The closure of a class of colimits” by M.H. Albert and G.M. Kelly, the authors were unable, except in one special case, to answer the following question. Let an indexing type ψ for colimits be such that every category admitting all ∅-indexed colimits for ∅ in some class Φ also admits ψ-indexed colimits; need a functor which preserves all Φ-indexed colimits also preserve ψ-indexed ones? Without settling the question, the present authors give two more positive results in this direction, including an affirmative answer when Φ-indexed colimits are conical.

The purpose of this paper is to study stable representations of partially ordered sets (posets) and compare it to the well known theory for quivers. In particular, we prove that every indecomposable representation of a poset of finite type is stable with respect to some weight and construct that weight explicitly in terms of the dimension vector. We show that if a poset is primitive then Coxeter transformations preserve stable representations. When the base field is the field of complex numbers we establish the connection between the polystable representations and the unitary $\chi$-representations of posets. This connection explains the similarity of the results obtained in the series of papers., Comment: Updated version. Appendix A was rewritten. Some corrections following the referee reports

The study of algebraic properties of ordered structures has shown that their behavior in many cases is different from algebraic structures . For example, the analogues of the fundamental mapping theorem for sets which characterizes surjective maps as quotient sets modulo their kernel relations, is not true for order-preserving maps between posets (partially ordered sets). The main objective of this paper is to study the quotients of dcpos (directed complete partially ordered sets), and their relations with surjective dcpo maps (directed join preserving maps). The motivation of studying such infinitary ordered structures is their importance in domain theory, a theory on the borderline of mathematics and theoretical computer science. In this paper, introducing the notion of a pre-congruence on dcpos (directed complete partially ordered sets), we give a characterization of dcpo congruences. Also, it is proved that unlike natural dcpo congruences, the dcpo congruences are precisely kernels of surjective dcpo maps. Also, while it is known that the image of a dcpo map is not necessarily a subdcpo of its codomain , we find equivalent conditions on a dcpo map to satisfy this property . Moreover, we prove the Decomposition Theorem and its consequences for dcpo maps.

The authors are grateful to George Janelidze for fruitful discussions on the topic of the present paper, and to the referee for several suggestions that helped to improve the final version of the paper. The first author was partially supported by Centro de Matemática da Universidade de Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020, and by the FCT Sabbatical Grant SFRH/BSAB/127925/2016. The second author was partially supported by the Fundação para a Ciência e a Tecnologia (FCT) and Centro2020 through the Project references: UID/Multi/04044/2013 and PAMI – ROTEIRO/0328/2013 (N.022158). The third author was partially supported by the Programma per Giovani Ricercatori “Rita Levi-Montalcini”, funded by the Italian government through MIUR. We study the categorical properties of preordered groups. We first give a description of limits and colimits in this category, and study some classical exactness properties. Then we point out a strong analogy between the algebraic behaviour of preordered groups and monoids, and we apply two different recent approaches to relative categorical algebra to obtain some homological properties of preordered groups. info:eu-repo/semantics/publishedVersion

We make several corrections and improvements to the published paper “Combinatorial categorical equivalences of Dold–Kan type”, mostly relating to the standing assumptions of the paper. In particular we have had to add one new assumption, but have been able to remove another.

The present paper is a sequel to our previous work on almost uniserial rings and modules, which appeared in the Journal of Algebra in 2016; it studies rings over which every (left and right) module is almost serial. A module is almost uniserial if any two of its submodules are either comparable in inclusion or isomorphic. And a module is almost serial if it is a direct sum of almost uniserial modules. The results of the paper are inspired by a characterization of Artinian serial rings as rings having all left (or right) modules serial. We prove that if R is a local ring and all left R-modules are almost serial then R is an Artinian ring which is uniserial either on the left or on the right. We also produce a connection between local rings having all left and right modules almost serial, local balanced rings studied by Dlab and Ringel and local Kothe rings. Finally we prove Morita invariance of the almost serial property and list some consequences.

In this paper, we study the Galois action on the extended Bloch groups of biquadratic and dihedral number fields. We prove that if F is a biquadratic number field, then the index Q 2 ( F ) in Browkin and Gangl's formulas on the Brauer–Kuroda relation can only be 1 or 2. This is exactly what Browkin and Gangl predicted in their paper. Moreover we give the explicit criteria for Q 2 ( F ) = 1 or 2 in terms of the Tate kernels. We also prove that Q 2 ( F ) = 1 or p for any dihedral extension F / Q whose Galois group is the dihedral group of order 2 p , where p is an odd prime.

In this paper, we introduce the notion of pseudo-absorbing multiplication modules. The purpose of this paper is to outline an approach to the classification of indecomposable pseudo-absorbing multiplication modules with finite-dimensional top over certain kinds of pullback rings and establish a connection between the pseudo-absorbing multiplication modules and the pure-injective modules over such rings.

We introduce the notion of a “category with path objects”, as a slight strengthening of Kenneth Brown's classical notion of a “category of fibrant objects”. We develop the basic properties of such a category and its associated homotopy category. Subsequently, we show how the exact completion of this homotopy category can be obtained as the homotopy category associated to a larger category with path objects, obtained by freely adjoining certain homotopy quotients. In a second part of this paper, we will present an application to models of constructive set theory. Although our work is partly motivated by recent developments in homotopy type theory, this paper is written purely in the language of homotopy theory and category theory, and we do not presuppose any familiarity with type theory on the side of the reader.

Given a graded monoid A with 1, one can construct a projective monoid scheme MProj ( A ) analogous to Proj ( R ) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves on MProj ( A ) , and we prove several basic results regarding these. We show that: 1. every quasicoherent sheaf F on MProj ( A ) can be constructed from a graded A-set in analogy with the construction of quasicoherent sheaves on Proj ( R ) from graded R-modules 2. if F is coherent on MProj ( A ) , then F ( n ) is globally generated for large enough n, and consequently, that F is a quotient of a finite direct sum of invertible sheaves 3. if F is coherent on MProj ( A ) , then Γ ( MProj ( A ) , F ) is finitely generated over A 0 (and hence a finite set if A 0 = { 0 , 1 } ). The last part of the paper is devoted to classifying coherent sheaves on P 1 in terms of certain directed graphs and gluing data. The classification of these over F 1 is shown to be much richer and combinatorially interesting than in the case of ordinary P 1 , and several new phenomena emerge.

Let F q be the finite field with q elements, p = char ( F q ) . The group GL 2 ( F q ) acts naturally in the set of irreducible polynomials over F q of degree at least 2. In this paper we are interested in the characterization and number of the irreducible polynomials that are fixed by the elements of a subgroup H of GL 2 ( F q ) . We make a complete characterization of the fixed polynomials in the case when H has only elements of the form ( 1 b 0 1 ) , corresponding to translations x ↦ x + b and, as a consequence, the case when H is a p-subgroup of GL 2 ( F q ) . This paper also contains alternative solutions for the cases when H is generated by an element of the form ( a 0 0 1 ) , obtained by Garefalakis (2010) and H = PGL 2 ( F q ) , obtained by Stichtenoth and Topuzoglu (2011).

Let Clt(A) denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping Clt(A)→Clt(A〚X〛), is injective and if A is a regular UFD, then Clt(A)→Clt(A〚X〛), is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping Clt(A)→Clt(A〚X〛); [I]↦[(I.A〚X〛)t] is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for Clt(A)→Clt(A〚X〛), to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an s∈S and a∈I such that sI⊆aA⊆I. The S-class group of A, S-Clt(A) is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-Clt(A)≃S-Clt(A〚X〛).

In a previous paper we generalized the theory of W*-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak*-rigged modules. At that time we promised a forthcoming paper devoted to other aspects of the theory. We fulfill this promise in the present work and its sequel "Rigged modules II", giving many new results about weak*-rigged modules and their tensor products. We also discuss the Picard group of weak* closed subalgebras of a commutative algebra. For example, we compute the weak Picard group of $H^{\infty}(D)$, and prove that for a weak* closed function algebra A, the weak Picard group of A is a semidirect product of the automorphism group of A, and the subgroup consisting of symmetric equivalence bimodules., 11 pages

A code algebra $A_C$ is a nonassociative commutative algebra defined via a binary linear code $C$. In a previous paper, we classified when code algebras are $\mathbb{Z}_2$-graded axial (decomposition) algebras generated by small idempotents. In this paper, for each algebra in our classification, we obtain the Miyamoto group associated to the grading. We also show that the code algebra structure can be recovered from the axial decomposition algebra structure., Comment: 25 pages

Let F be a field of characteristic 2. The aim of this paper is to give a complete proof of the norm theorem for singular F-quadratic forms which are not totally singular, i.e., we give necessary and sufficient conditions for which a normed irreducible polynomial of F [ x 1 , … , x n ] becomes a norm of such a quadratic form over the rational function field F ( x 1 , … , x n ) . This completes partial results proved on this question in [8] . Combining the present work with the papers [1] and [7] , we obtain the norm theorem for any type of quadratic forms in characteristic 2.

Several papers deal with the problem of counting the number of idempotent endomorphisms of a structure S onto a substructure T. In this paper we consider the case when T is a projective lattice-ordered abelian group with a distinguished strong order unit, or equivalently, a projective MV-algebra. Suppose A is the image (=range) of an idempotent endomorphism of the free n-generator MV-algebra M ( [ 0 , 1 ] n ) of McNaughton functions on [ 0 , 1 ] n . We prove that the number r ( A ) of idempotent endomorphisms of M ( [ 0 , 1 ] n ) onto A is finite if, and only if, the maximal spectral space μ A is homeomorphic to a (Kuratowski) closed domain M of [ 0 , 1 ] n , in the sense that M = cl ( int ( M ) ) . Further, the closed domain condition is decidable and r ( A ) is computable, once an idempotent endomorphism of M ( [ 0 , 1 ] n ) onto A is explicitly given. Thus every finitely generated projective MV-algebra B comes equipped with a new invariant ι ( B ) = sup ⁡ { r ( A ) | A ≅ B , for A the image of an idempotent endomorphism of M ( [ 0 , 1 ] k ) } , and k the smallest number of generators of B. We compute ι ( B ) for many projective MV-algebras B existing in the literature. Various problems concerning idempotent endomorphisms of free MV-algebras are shown to be decidable. Via the Γ functor, our results and computations automatically transfer to finitely generated projective abelian l-groups with a distinguished strong unit.

We use the terms ∞-categories and ∞-functors to mean the objects and morphisms in an ∞-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects. Quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, θ n -spaces, and fibered versions of each of these are all ∞-categories in this sense. Previous work in this series shows that the basic category theory of ∞-categories and ∞-functors can be developed only in reference to the axioms of an ∞-cosmos; indeed, most of the work is internal to the homotopy 2-category, a strict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi-categories, we recapture precisely the same category theory developed by Joyal and Lurie, although our definitions are 2-categorical in natural, making no use of the combinatorial details that differentiate each model. In this paper, we introduce cartesian fibrations, a certain class of ∞-functors, and their groupoidal variants. Cartesian fibrations form a cornerstone in the abstract treatment of “category-like” structures a la Street and play an important role in Lurie's work on quasi-categories. After setting up their basic theory, we state and prove the Yoneda lemma, which has the form of an equivalence between the quasi-category of maps out of a representable fibration and the quasi-category underlying the fiber over its representing element. A companion paper will apply these results to establish a calculus of modules between ∞-categories, which will be used to define and study pointwise Kan extensions along ∞-functors.

This is the first of a series of papers on a categorical approach to the logical systems. Its aim is to set forth the necessary foundations for more advanced concepts. The paper shows how the internal models of various lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting two famous theorems: one saying that under some mild conditions, an object is internally complete iff it is internally cocomplete; and another saying that an object of a sufficiently cocomplete 2-category cannot be internally (co)complete unless it is degenerated. The first of the theorems specialises to a well-known fact from order theory, and also provides non-trivial results about posets and categories in constructive mathematics. Second of the theorems gives a powerful generalisation of Freyd's Theorem and sheds more light on the difficulty of finding fibrational models for higher polymorphism. As a simple corollary of this theorem, we obtain a variant of Freyd's Theorem for the categories internal to any tensored category. There is also a hidden objective of the paper — reading it backwards should provide a gentle introduction to the 2-categorical concepts through internal categories.

This paper is a further contribution to the developing theory of Boolean inverse monoids. These monoids should be regarded as non-commutative generalizations of Boolean algebras; indeed, classical Stone duality can be generalized to this non-commutative setting to yield a duality between Boolean inverse monoids and a class of etale topological groupoids. MV-algebras are also generalizations of Boolean algebras which arise from many-valued logics. It is the goal of this paper to show how these two generalizations are connected. To do this, we define a special class of Boolean inverse monoids having the property that their lattices of principal ideals naturally form an MV-algebra. We say that an arbitrary MV-algebra can be co-ordinatized if it is isomorphic to an MV-algebra arising in this way. Our main theorem is that every countable MV-algebra can be so co-ordinatized. The particular Boolean inverse monoids needed to establish this result are examples of what we term AF inverse monoids and are the inverse monoid analogues of AF C ⁎ -algebras. In particular, they are constructed from Bratteli diagrams as direct limits of finite direct products of finite symmetric inverse monoids.

Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R -polynomials for the symmetric group. Let SnSn be the symmetric group on {1,2,…,n}{1,2,…,n}, and let S={si|1≤i≤n−1} be the generating set of SnSn, where for 1≤i≤n−11≤i≤n−1, sisi is the adjacent transposition. For a subset J⊆SJ⊆S, let (Sn)J(Sn)J be the parabolic subgroup generated by J , and let (Sn)J(Sn)J be the set of minimal coset representatives for Sn/(Sn)JSn/(Sn)J. For u≤v∈(Sn)Ju≤v∈(Sn)J in the Bruhat order and x∈{q,−1}x∈{q,−1}, let Ru,vJ,x(q) denote the parabolic R-polynomial indexed by u and v . Brenti found a formula for Ru,vJ,x(q) when J=S∖{si}J=S∖{si}, and obtained an expression for Ru,vJ,x(q) when J=S∖{si−1,si}J=S∖{si−1,si}. In this paper, we provide a formula for Ru,vJ,x(q), where J=S∖{si−2,si−1,si}J=S∖{si−2,si−1,si} and i appears after i−1i−1 in v. It should be noted that the condition that i appears after i−1i−1 in v is equivalent to that v is a permutation in (Sn)S∖{si−2,si}(Sn)S∖{si−2,si}. We also pose a conjecture for Ru,vJ,x(q), where J=S∖{sk,sk+1,…,si}J=S∖{sk,sk+1,…,si} with 1≤k≤i≤n−11≤k≤i≤n−1 and v is a permutation in (Sn)S∖{sk,si}(Sn)S∖{sk,si}.

We show that if k is an infinite field, then there exists a subspace W⊂kN of dimension |k|ℵ0, such that no nonzero member of W has infinitely many zeros. This generalizes a result from a paper by Bergman and Nahlus, and partly answers another question from the same paper.

In the first half of this paper, we develop the fundamentals of Lie–Poisson theory for direct limits G = lim → G n of complex algebraic groups and their Lie algebras g = lim → g n . We describe the Poisson pro- and ind-variety structures on g ⁎ = lim ← g n ⁎ and the coadjoint orbits of G , respectively. While the existence of symplectic foliations remains an open question for most infinite-dimensional Poisson manifolds, we show that for direct limit algebras, the coadjoint orbits give a weak symplectic foliation of the Poisson provariety g ⁎ . The second half of the paper applies our general results to the concrete setting of G = GL ( ∞ ) and g ⁎ = M ( ∞ ) , the space of infinite-by-infinite complex matrices with arbitrary entries. We use the Poisson structure of g ⁎ to construct an integrable system on M ( ∞ ) that generalizes the Gelfand–Zeitlin system on gl ( n , C ) to the infinite-dimensional setting. We further show that this integrable system integrates to a global action of a direct limit group on M ( ∞ ) , whose generic orbits are Lagrangian ind-subvarieties of the coadjoint orbits of GL ( ∞ ) on M ( ∞ ) .

A commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains, noting the almost local nature of the property and, in the case of a local domain, relating it to the structure of its integral closure. In a 2011 paper, Ay and Klingler obtain similar results for Noetherian reduced rings. In this paper, we examine the UDI property for arbitrary commutative Noetherian rings, establishing the same almost local nature of the property, and giving an example which shows that the local results do not extend to commutative Noetherian rings in general.

We consider a certain subgroup A n + of the automorphism group of a free group of rank n. It can be regarded as a free group analogue of the group Λ n of integral lower-triangular matrices. We call A n + the lower-triangular automorphism group of a free group. The first aim of the paper is to give a finite presentation for A n + . The abelianization of the free group induces the surjective homomorphism ρ + from A n + to Λ n . In our previous paper [18] , we introduced the lower-triangular IA-automorphism group IA n + . Here we show that IA n + coincides with the kernel of ρ + . The second aim of the paper is to give an infinite presentation for IA n + . Finally, we study a relation of the second homology groups between A n + and Λ n . In particular, we compute the second homology group H 2 ( Λ n , L ) by using Magnus's presentation where L is a principal ideal domain in which 2 is invertible. For example, L = Q , Z / p Z for prime p ≥ 3 . This gives a lower bound on the integral second homology group of A n + .

Let K = Q ( θ ) be an algebraic number field with θ in the ring A K of algebraic integers of K having minimal polynomial f ( x ) over Q . For a prime number p, let i p ( f ) denote the highest power of p dividing the index [ A K : Z [ θ ] ] . Let f ¯ ( x ) = ϕ ¯ 1 ( x ) e 1 ⋯ ϕ ¯ r ( x ) e r be the factorization of f ( x ) modulo p into a product of powers of distinct irreducible polynomials over Z / p Z with ϕ i ( x ) ∈ Z [ x ] monic. Let the integer l ≥ 1 and the polynomial N ( x ) ∈ Z [ x ] be defined by f ( x ) = ∏ i = 1 r ϕ i ( x ) e i + p l N ( x ) , N ‾ ( x ) ≠ 0 ¯ . In this paper, we prove that i p ( f ) ≥ ∑ i = 1 r u i deg ⁡ ϕ i ( x ) , where u i is a constant defined only in terms of l , e i and the highest power of the polynomial ϕ ¯ i ( x ) dividing N ‾ ( x ) . Further a class of irreducible polynomials is described for which the above inequality becomes equality. The results of the paper quickly yield the well known Dedekind criterion which gives a necessary and sufficient condition for i p ( f ) to be zero. In fact, these results are proved in a more general set up replacing Z by any Dedekind domain.

This is one of two papers on the injective spectrum of a right noetherian ring. In [4] , we defined the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples. In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure sheaf and their relation to modules over the original ring. We then explore links with the spectrum of prime torsion theories developed by Golan [2] and use this torsion-theoretic viewpoint to prove further results about the topology.

This paper studies finitely generated quasivarieties of Sugihara algebras. These quasivarieties provide complete algebraic semantics for certain propositional logics associated with the relevant logic R-mingle. The motivation for the paper comes from the study of admissible rules. Recent earlier work by the present authors, jointly with Freisberg and Metcalfe, laid the theoretical foundations for a feasible approach to this problem for a range of logics—the Test Spaces Method. The method, based on natural duality theory, provides an algorithm to obtain the algebra of minimum size on which admissibility of sets of rules can be tested. (In the most general case a set of such algebras may be needed rather than just one.) The method enables us to identify this ‘admissibility algebra’ for each quasivariety of Sugihara algebras which is generated by an algebra whose underlying lattice is a finite chain. To achieve our goals, it was first necessary to develop a (strong) duality for each of these quasivarieties. The dualities promise also to provide a valuable new tool for studying the structure of Sugihara algebras more widely.

We give an a priori proof of the known presentations of (that is, completeness of families of relations for) the principal subspaces of all the standard A 1 ( 1 ) -modules. These presentations had been used by Capparelli, Lepowsky and Milas for the purpose of obtaining the classical Rogers–Selberg recursions for the graded dimensions of the principal subspaces. This paper generalizes our previous paper.

In a previous paper (Pure Appl. Algebra 159 (2001) 231) a definition of point of a Gelfand quantale is given in terms of algebraically irreducible representations of the quantale on an atomic orthocomplemented sup-lattice. The definition yields the usual notion of point when applied to a locale viewed as a quantale and helps to establish that the spectrum Max A of a C ∗ -algebra A is an invariant of A. The current paper is concerned with finding a notion of spatiality in which, as intuitively expected, Max A is spatial, and for which there is an intimate connection with points in the above sense. Moreover, the approach taken relates naturally to that of Giles and Kummer in their article on non-commutative spaces (Indiana Math. J. 21 (1971) 91). Explicitly, an involutive unital quantale X is said to be spatial provided that it admits an algebraically strong right embedding into a discrete von Neumann quantale Q. It is proved that X is spatial if, and only if, it has enough points, and that, in this case, it is necessarily a Gelfand quantale. In particular, a locale is spatial as an involutive unital quantale precisely when it yields the topology of a classical topological space. A notion of quantal space is defined, and it is shown that any involutive unital quantale admits a spatialisation determined by its points, generalising that known for locales. In particular, any quantal space admits an underlying topological space. Finally, Max A is shown to be spatial, and to determine a canonical quantal space, as desired.

The present second part of a three-part paper gives the detailed treatment of the new notion of multicategory, and that of the construction of the particular multicategory of function replacement. For the overall purpose of the whole paper, see the abstract in Part 1.

This paper concerns three classes of geometric 2-complexes of nonpositive curvature: one in which all of the 2-cells are squares, one in which they are all equilateral triangles, and one in which they are all regular hexagons. (These cases correspond to the three regular tessellations of the euclidean plane.) These three classes of 2-complexes, while highly restrictive, are nevertheless useful because they include the Cayley complexes of group presentations satisfying the small cancellation conditions C″(p)− T (q) for (p,q)∈{(3,6),(4,4),(6,3)} (and satisfying the additional condition that all the relators have length exactly p ). These three cases are of particular interest because such groups are not necessarily word-hyperbolic in the sense of Gromov (all other classes of C″(p)− T (q) groups are, see S.M. Gersten, H. Short, Small cancellation theory and automatic groups, Invent. Math. 102 (1990) 305–334). Moreover, they include presentations of alternating link groups (under free product with the infinite cyclic group; see C. Weinbaum, The word and conjugacy problem for the knot group of any tame prime alternating knot, Proc. Math. Soc. 22 (1971) 22–26). This paper is a study of geodesic edge paths in the geometric complexes described above, and consequently, of geodesic words in C″(p)− T (q) group presentations. The most fundamental theorem on this topic, namely that an edge path in the complex (or equivalently, a word on the group generators) that fails to be geodesic can be replaced via an efficient algorithm with an equivalent one that is geodesic, is well-known; but careful proofs of this fact, as well as related facts, are difficult or impossible to find in the literature. This paper is intended to provide rigorous proofs of some useful facts about the geodesics in these complexes and group presentations. Some embedding and minimality sufficiency conditions for disks are shown, together with an algorithm for finding all geodesics in an equivalence class. Some useful implications are mentioned K. Johnsgard, Two automatic spanning trees in small cancellation group presentations, Int. J. Algebra. Comput. 6 (1994) 429–440, K. Johnsgard, The conjugacy problem for groups of alternating prime tame links is polynomial-time, Trans. Amer. Math. Soc. 349 (1997) 847–901.

In a collection of classical papers, R. Nunke studied radicals R on the category of abelian groups constructed using extensions of the integers. In particular, the sequences in RExtAG were called R-pure. In this paper several types of purity related to R-purity are discussed. Projective and injective resolutions of an arbitrary group are constructed, providing interesting examples of dual classes of groups. One particular variation, that of R∗-purity, is shown to have many of the advantages of R-purity, while suffering from fewer of its drawbacks, e.g., every group has an R∗-injective hull and the class of R∗-projectives is closed under arbitrary subgroups. The most important example is R=pα, where α is an ordinal. When α is a limit, pα-purity and its generalizations are closely related to the completion of groups in the α-topology. Although these are the motivating examples, many of the results are stated in a substantially more general context.

The fact that a real univariate polynomial misses some real roots is usually overcame by considering complex roots, but the price to pay for, is a complete lost of the sign structure that a set of real roots is endowed with (mutual position on the line, signs of the derivatives, etc...). In this paper we present real substitutes for these missing roots which keep sign properties and which extend of course the existing roots. Moreover these "virtual roots" are the values of semialgebraic continuous -- rather uniformly -- functions defined on the set of monic polynomials. We present some applications., Comment: tiny corrections to the original paper

In a previous paper the last two authors introduced a condition which gave an elementwise characterization of subintegrality for an extension A ⊆ B of commutative Q-algebras. In the present paper we show that the same condition gives an elementwise characterization of weak subintegrality for an extension A ⊆ B of arbitrary commutative rings. We also give a new characterization of weakly subintegral elements in which the “coefficients” lie in A rather than B.

Twisted vertex operators based on rational lattices have had many applications in vertex operator algebra theory and conformal field theory. In this paper, ``relativized'' twisted vertex operators are constructed in a general context based on isometries of rational lattices, and a generalized twisted Jacobi identity is established for them. This result generalizes many previous results. Relatived untwisted vertex operators had been studied in a monograph by the authors. The present paper includes as a special case the proof of the main relations among twisted vertex operators based on even lattices announced some time ago by the second author., Latex, 38 pages, to appear in J. Pure & Applied Algebra

This is the fourth part of a series of papers developing a tensor product theory of modules for a vertex operator algebra. In this paper, We establish the associativity of $P(z)$-tensor products for nonzero complex numbers $z$ constructed in Part III of the present series under suitable conditions. The associativity isomorphisms constructed in this paper are analogous to associativity isomorphisms for vector space tensor products in the sense that it relates the tensor products of three elements in three modules taken in different ways. The main new feature is that they are controlled by the decompositions of certain spheres with four punctures into spheres with three punctures using a sewing operation. We also show that under certain conditions, the existence of the associativity isomorphisms is equivalent to the associativity (or (nonmeromorphic) operator product expansion in the language of physicists) for the intertwining operators (or chiral vertex operators). Thus the associativity of tensor products provides a means to establish the (nonmeromorphic) operator product expansion., Comment: LaTeX file. 54 pages. To appear in the special 100th issue of the Journal of Pure and Applied algebra, dedicated to ``Applications of Algebra to Physics.''

One of our main results is a classification of all the possible quivers of selfinjective radical cube zero finite-dimensional algebras over an algebraically closed field having finite complexity. In the paper (Erdmann and Solberg, 2011) [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite-dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5] .

Introduction The connection between ∗-autonomous categories and linear logic has been pointed out in several places, including [Seely, 1989] and [Barr, to appear]. The only essential difference between them is the question of the existence of cofree coalgebras (counitary, coassociative, cocommutative, to be precise). In my previous paper, I gave a construction that did lead to models of linear logic, but only by beginning with a category that was already cartesian closed (with additional properties). This paper follows quite a different tack. We begin with a category that is locally presentable in the sense of [Gabriel & Ulmer, 1971], which is equivalent to its being accessible in the sense of [Makkai & Pare, 1990] together with being either complete or cocomplete, as well as being autonomous and show that in that case the ∗-autonomous category constructed in [Chu, 1979] has cofree coalgebras and is thus a model of full linear logic.

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α -abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we study crossed modules of Hom-Lie algebras showing their equivalence with cat 1 -Hom-Lie algebras, and we introduce α -crossed modules to have a better understanding of the third cohomology group.

In this paper, a holomorphic vertex operator algebra U of central charge 24 with the weight one Lie algebra A 8 , 3 A 2 , 1 2 is proved to be unique. Moreover, a holomorphic vertex operator algebra of central charge 24 with the weight one Lie algebra F 4 , 6 A 2 , 2 is obtained by applying a Z 2 -orbifold construction to U. The uniqueness of such a vertex operator algebra is also established. By a similar method, we also established the uniqueness of a holomorphic vertex operator algebra of central charge 24 with the weight one Lie algebra E 7 , 3 A 5 , 1 . As a consequence, we verify that all 71 Lie algebras in Schellekens' list can be realized as the weight one Lie algebras of some holomorphic vertex operator algebras of central charge 24. In addition, we establish the uniqueness of three holomorphic vertex operator algebras of central charge 24 whose weight one Lie algebras have the type A 8 , 3 A 2 , 1 2 , F 4 , 6 A 2 , 2 , and E 7 , 3 A 5 , 1 .

The goal of this paper is to address the problem of building a path object for the category of Grothendieck (weak) ∞-groupoids. This is the missing piece for a proof of Grothendieck's homotopy hypothesis. We show how to endow the putative underlying globular set with a system of composition, a system of identities and a system of inverses, together with an approximation of the interpretation of any map for a theory of ∞-categories. Finally, we introduce a coglobular ∞-groupoid representing modifications of ∞-groupoids, and prove some basic properties it satisfies, that will be exploited to interpret all 2-dimensional categorical operations on cells of the path object P X of a given ∞-groupoid X .

© 2019 Elsevier B.V. In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds for all smooth curves of genus at most 32 or Clifford index at most 6 on arbitrary toric surfaces. Conversely we use known results on Green's conjecture (due to Lelli-Chiesa)to obtain new facts about graded Betti tables of projectively embedded toric surfaces. ispartof: JOURNAL OF PURE AND APPLIED ALGEBRA vol:224 issue:2 pages:507-527 status: published

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form., 22 pages; v3 minor corrections/clarifications; to appear in JPAA

In this paper we develop a descent theory for morphisms α between a monoid B and a unital magma A in a monoidal category with equalizers and coequalizers. We introduce the category of strong descent data for α and we prove that under faithfully flat conditions this category is equivalent to the one of right B-modules. As an application we prove that the category of strong Hopf modules, introduced by us for Hopf quasigroups and weak Hopf quasigroups, is equivalent to a suitable category of strong descent data.

As a first objective, we characterise those essentially algebraic categories which satisfy properties like being unital, strongly unital, n-permutable, subtractive or protomodular. For each such property, we obtain a Mal'tsev condition as an equivalent condition. Using the language of Janelidze matrix conditions, we treat many of these properties together. As a second objective, using these characterisations, we prove some embedding theorems for those properties in a regular context in the same style as we did in the companion paper [23] . Concrete examples of how to use these embedding theorems are given. Finally, to extend those embedding theorems to the exact context, we show that these properties are stable under the exact completion of a regular category.

Let $R=K[X_1,\dots, X_n]$ be a polynomial ring in $n$ variables over a field $K$ of charactersitic zero and $d$ a $K$-derivation of $R$. Consider the isotropy group if $d$: $ \text{Aut}(R)_d :=\{\rho \in \text{Aut}_K(R)|\; \rho d \rho^{-1}=d\}$. In his doctoral thesis, Baltazar proved that if $d$ is a simple Shamsuddin derivation of $K[X_1,X_2]$, then its isotropy group is trivial. He also gave an example of a non-simple derivation whose isotropy group is infinite. Recently, Mendes and Pan generalized this result to an arbitrary derivation of $K[X_1,X_2]$ proving that a derivation of $K[X_1,X_2]$ is simple if, and only if, its isotropy group is trivial. In this paper, we prove that the isotropy group of a simple Shamsuddin derivation of the polynomial ring $R=K[X_1,\dots, X_n]$ is trivial. We also calculate other isotropy groups of (not necessarily simple) derivations of $K[X_1,X_2]$ and prove that they are finite cyclic groups., Comment: 8 pages

In this paper, we characterize the (generalized) Frobenius powers and critical exponents of two classes of monomial ideals of a polynomial ring in positive characteristic: powers of the homogeneous maximal ideal, and ideals generated by positive powers of the variables. In doing so, we effectively characterize the test ideals and $F$-jumping exponents of sufficiently general homogeneous polynomials, and of all diagonal polynomials. Our characterizations make these invariants computable, and show that they vary uniformly with the congruence class of the characteristic modulo a fixed integer. Moreover, we confirm that for a diagonal polynomial over a field of characteristic zero, the test ideals of its reduction modulo a prime agree with the reductions of its multiplier ideals for infinitely many primes., Comment: 19 pages, 1 figure, comments welcome

In this paper, we consider automorphism groups of rational surfaces which admit cuspidal anticanonical curves and have certain nontrivial automorphisms. By applying Coxeter theory, we show that the automorphism groups of the surfaces are isomorphic to the infinite cyclic group.

In the past few years, an action of the group PGL 2 ( F q ) on the set of monic irreducible polynomials over F q has been introduced and questions on the characterization and number of fixed points were discussed. In this paper we show that, in general, such fixed points arise from rational transformations, extending a work of Mattarei and Pizzato (2017).