Narrow Results

The category ${STROP}_m$ of supertropical monoids, whose morphisms are transmissions, has the full-reflective subcategory $STROP$ of commutative semirings. In this setup, quotients are determined directly by equivalence relations, as ideals are not applicable for monoids, leading to a new approach to factorization theory. To this end, tangible factorization into irreducibles is obtained through fiber contractions and their hierarchy. Fiber contractions also provide different quotient structures, associated with covers and types of splitting covers.

We derive the equivalent energy of a square lattice that either deforms into the three- dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy–Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomistic models.

This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18.

Let be a Kac–Moody Lie algebra. We give an interpretation of Tits' associated group functor using representation theory of and we construct a locally compact "Kac–Moody group" G over a finite field k. Using (twin) BN-pairs (G,B,N) and (G,B^-,N) for G we show that if k is "sufficiently large", then the subgroup B^- is a non-uniform lattice in G. We have also constructed an uncountably infinite family of both uniform and non-uniform lattices in rank 2. We conjecture that these form uncountably many distinct conjugacy classes in G. The basic tool for the construction of non-uniform lattices in rank 2 is a spherical Tits system for G which we also construct.

We continue the study of the preradicals of a ring in the lattice point of view. We introduce several interesting preradicals associated to a given preradical and some partitions of the whole lattice in terms of preradicals. As an application, we also give some classification theorems.

In this paper we study prime preradicals, irreducible preradicals, ∧-prime preradicals, prime submodules and diuniform modules. We study some relations between these concepts, using the lattice structure of preradicals developed in previous papers. In particular, we give a characterization of prime preradicals using an operator named the relative annihilator. We also characterize prime submodules by means of prime preradicals. We give some characterizations of rings that have certain conditions on prime radicals and on irreducible preradicals, such as left local left V-rings, as well as 1-spr rings, which we introduce.

Our main result states that a finite semiring of order > 2 with zero which is not a ring is congruence-simple if and only if it is isomorphic to a "dense" subsemiring of the endomorphism semiring of a finite idempotent commutative monoid.

We also investigate those subsemirings further, addressing e.g. the question of isomorphy.

Let A be a symmetrizable affine or hyperbolic generalized Cartan matrix. Let G be a locally compact Kac–Moody group associated to A over a finite field 𝔽_q. We suppose that G has type ∞, that is, the Weyl group W of G is a free product of ℤ/2ℤ's. This includes all locally compact Kac–Moody groups of rank 2 and three possible locally compact rank 3 Kac–Moody groups of noncompact hyperbolic type. For every prime power q, we give a sufficient condition for the rank 2 Kac–Moody group G to contain a cocompact lattice with quotient a simplex, and we show that this condition is satisfied when q = 2^s. If further M_q and are abelian, we give a method for constructing an infinite descending chain of cocompact lattices … Γ₃ ≤ Γ₂ ≤ Γ₁ ≤ Γ. This allows us to characterize each of the quotient graphs of groups Γ_i\\X, the presentations of the Γ_i and their covolumes, where X is the Tits building of G, a homogeneous tree. Our approach is to extend coverings of edge-indexed graphs to covering morphisms of graphs of groups with abelian groupings. This method is not specific to cocompact lattices in Kac–Moody groups and may be used to produce chains of subgroups acting on trees in a general setting. It follows that the lattices constructed in the rank 2 Kac–Moody group have the Haagerup property. When q = 2 and rank(G) = 3 we show that G contains a cocompact lattice Γ′₁ that acts discretely and cocompactly on a simplicial tree . The tree is naturally embedded in the Tits building X of G, a rank 3 hyperbolic building. Moreover Γ′₁ ≤ Λ′ for a non-discrete subgroup Λ′ ≤ G whose quotient Λ′ \ X is equal to G\X. Using the action of Γ′₁ on we construct an infinite descending chain of cocompact lattices …Γ′₃ ≤ Γ′₂ ≤ Γ′₁ in G. We also determine the quotient graphs of groups , the presentations of the Γ′_i and their covolumes.

This article consists of two sections. In the first one, the concepts of spanning and cospanning classes of modules, both hereditarily and cohereditarily, are explained, and some closure properties of the class of modules hereditarily cospanned by a conatural class are established, which amount to its being a hereditary torsion class. This gives a function from R-conat to R-tors and it is proven that its being a lattice isomorphism is part of a characterization of bilaterally perfect rings. The second section begins considering a description of pseudocomplements in certain lattices of module classes. The idea is generalized to define an inclusion-reversing operation on the collection of classes of modules. Restricted to R-nat, it is shown to be a function onto R-tors, and its being an anti-isomorphism is equivalent to R being left semiartinian. Lastly, another characterization of R being left semiartinian is given, in terms solely of R-tors.

Given a complete modular meet-continuous lattice $A$, an inflator on $A$ is a monotone function $d:A\to A$ such that $a\le d(a)$ for all $a\in A$. If $I(A)$ is the set of all inflators on $A$, then $I(A)$ is a complete lattice. Motivated by preradical theory, we introduce two operators, the totalizer and the equalizer. We obtain some properties of these operators and see how they are related to the structure of the lattice $A$ and with the concept of dimension.

The main purpose of this work is to give a constructive proof for a particular case of the no-name lemma. Let $G$ be a finite group, $K$ a field that is equipped with a faithful $G$-action, and $L$ a sign permutation $G$-lattice (see the Introduction for the definition). Then $G$ acts naturally on the group algebra $K[L]$ of $L$ over $K$, and hence also on the quotient field $K(L)=Q(K[L])$. A well-known variant of the no-name lemma asserts that the invariant sub-field $K{(L)}^G$ is a purely transcendental extension of $K^G$. In other words, there exist $y_1,\dots ,y_n$ which are algebraically independent over $K^G$ such that $K{(L)}^G\cong K^G(y_1,\dots ,y_n)$. In this paper, we give an explicit construction of suitable elements $y_1,\dots ,y_n$.

In this paper, given an arbitrary set $\mathrm{\Omega }$, we study the main order and algebraic properties of some maps and set structures that are strictly related to dependence set relations on $\mathrm{\Omega }$, which are binary relations between subsets of $\mathrm{\Omega }$ naturally arising when $\mathrm{\Omega }$ is a topological space or an attribute set in rough set theory and granular computing based on information systems. The previous maps, that we call granular maps, have the families of the set systems, set operators, binary set relations or also of information systems on the ground set $\mathrm{\Omega }$ as their domain and codomain. We make use of various algebraic methodologies on granular maps to determine the main order-theoretic and combinatorial properties of specific sub-collections of set systems, binary set relations and set operators naturally arising in the investigation of dependence set relations and of rough set theory. We introduce, in more detail, the notion of granular sub-bijection to formalize in all these situations the undefined notion of cryptomorphism, and through which we exhibit new equivalences between specific families of set systems, binary set relations and set operators strictly related to dependence set relations. By means of suitable granular maps we determine three granular sub-bijections between the family of all the closure operators, that of all the Moore set systems and that of all dependence set relations on the same ground set $\mathrm{\Omega }$. Next, through a property of adjunctivity, we see that in order to generate a dependence set relation it suffices to consider pointed relations on $\mathrm{\Omega }$, namely collections of pairs in $\mathrm{\Omega }\times \wp (\mathrm{\Omega })$. Because of that, we study order-theoretical properties of some relevant subclasses of pointed relations and analyze the granular maps on $\mathrm{\Omega }$ which determine two nontrivial granular sub-bijections between two subclasses of set operators and two corresponding subclasses of pointed relations. Next, we show that any dependence set relation has the form ${\mathrm{\text{Dep}}}_{𝔓}$ that is a dependence set relation induced by an information system $𝔓$ on $\mathrm{\Omega }$ and generalizes the Pawlak dependence set relation frequently used in rough set theory. With regard to this representation result, we characterize some set systems of minimal subsets with respect to the Pawlak indiscernibility relation on information systems. Finally, given an arbitrary binary set relation ${𝒟}$ on $\mathrm{\Omega }$, we consider the smallest dependence set relation ${{𝒟}}^{+}$ on $\mathrm{\Omega }$ containing ${𝒟}$ and call it dependence closure of ${𝒟}$. Then, when $\mathrm{\Omega }$ is a finite set, we show how to generate ${{𝒟}}^{+}$ in four different and recursive ways by starting from ${𝒟}$. Moreover, again in the finite case, given an information system $𝔓$ on $\mathrm{\Omega }$, we also determine a binary set relation ${{ℒ}}_{𝔓}$ on $\mathrm{\Omega }$ for which ${{ℒ}}_{𝔓}^{+}$ agrees with ${\text{Dep}}_{𝔓}$ and whose cardinality is minimum with respect to that of all binary set relations whose dependence closure agrees with ${\text{Dep}}_{𝔓}$.

Let $V_L$ be the vertex algebra associated with a non-degenerate and non-positive definite even lattice $L$ and $V_L^{+}$ the fixed point subalgebra of $V_L$ under the action of the automorphism induced from the $-1$-isometry of $L$. We determine the fusion rules for weak $V_L^{+}$-modules. As an application of this result, we classify the irreducible weak modules for the fixed point subalgebra of $V_L$ under the action of the automorphism induced from an isometry of $L$ of order $2$. We also show that every weak module for the same fixed point subalgebra is completely reducible.