Modular Law Research Articles

Modular law through GKM theory

Modular law through GKM theory

On the lattice of closed subgroups of a profinite group

The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behavior of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained.

General Mixed Lattices

AbstractA mixed lattice is a lattice-type structure consisting of a set with two partial orderings, and generalizing the notion of a lattice. Mixed lattice theory has previously been studied in various algebraic structures, such as groups and semigroups, while the more general notion of a mixed lattice remains unexplored. In this paper, we study the fundamental properties of mixed lattices and the relationships between the various properties. In particular, we establish the equivalence of the one-sided associative, distributive and modular laws in mixed lattices. We also give an alternative definition of mixed lattices and mixed lattice groups as non-commutative and non-associative algebras satisfying a certain set of postulates. The algebraic and the order-theoretic definitions are then shown to be equivalent.

Chromatic symmetric functions from the modular law

In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced in [19]. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain q-hit numbers (up to a factor). Finally, we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function.

Equivalent conditions for modularity

In this paper, we give a list of identities which are equivalent to modular law. The identities are divided into various small groups for the convenience in proofs.

From modules to lattices: Insight into the genesis of Dedekind's Dualgruppen

When Dedekind introduced the notion of a module, he also defined their divisibility and related arithmetical notions (e.g. the LCM of modules). The introduction of notations for these notions allowed Dedekind to state new theorems, now recognized as the modular laws in lattice theory. Observing the dualism displayed by the theorems, Dedekind pursued his investigations on the matter. This led him, 20 years later, to introduce Dualgruppen, equivalent to lattices Dedekind [Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler. In Dedekind (1930–1932), volume II, 1897, 103–147; Über die von drei Moduln erzeugte Dualgruppe. In Dedekind (1930–1932), volume II, 1900, 236–271]. After a brief exposition of the basic elements of Dualgruppe theory, and with the help of his Nachlass, I show how Dedekind gradually built his theory through layers of computations, often repeated in slight variations and attempted generalizations. I study the tools he devised to help and accompany him in his computations. I highlight the crucial conceptual move that consisted in going from investigating operations between modules, to groups of modules closed under these operations. By using Dedekind's drafts, I aim to highlight the concealed yet essential practices anterior to the published text.

A characterization of modular law and distributive law of lattices by the Yang–Baxter maps

In this article, we generalize a bubble sort map to a map on the lattice, and characterize the modular law and the distributive law of lattices from the viewpoint of the Yang–Baxter maps.

On modular inequalities of interval-valued fuzzy soft sets characterized by soft J-inclusions

Abstract This study aims to explore modular inequalities of interval-valued fuzzy soft sets characterized by Jun’s soft J-inclusions. Using soft product operations of interval-valued fuzzy soft sets, we first investigate some basic properties of soft J-inclusions and soft L-inclusions. Then a new concept called upward directed interval-valued fuzzy soft sets is defined and some equivalent characterizations are presented. Furthermore, we consider modular laws in lattice theory and find that classical modular inequalities in lattice theory are not valid for interval-valued fuzzy soft sets. Finally, we present some interesting inequalities of interval-valued fuzzy soft sets by virtue of soft J-inclusions and related notions. MSC:03E72.

On Supplement Submodules

UDC 512.5 We investigate some properties of supplement submodules. Some relations between lying-above and weak supplement submodules are also studied. Let V be a supplement of a submodule U in M: Then it is possible to define a bijective map between the maximal submodules of V and the maximal submodules of M that contain U: Let M be an R -module, U M; let V be a weak supplement of U; and let K V: In this case, K is a weak supplement of U if and only if V lies above K in M: We prove that an R -module M is amply supplemented if and only if every submodule of M lies above a supplement in M: We also prove that M is semisimple if and only if every submodule of M is a supplement in M: Throughout this paper R is an arbitrary ring with identity and all modules are unital left R-modules. Let M be an R-module and let V be a submodule of M: If L = M for every submodule L of M such that V +L = M; then V is called a small submodule of M; and we write V << M: In the present work, we denote the intersection of all maximal submodules of M by Rad(M): If M has no maximal submodules, then we define Rad(M) = M: Let M be an R-module. We write N M to denote that N is a submodule of M: Lemma 1.1 (Modular law). Let M be an R-module, let K; N; and H be submodules of M; and let H N: Then N\ (H + K) = H + N\ K (see [3]). Let U be a submodule of an R-module M . If a submodule V is minimal in the collection of submodules L of M such that U +L = M; then V is called a supplement of U by addition or simply a supplement of U in M: In this case, it is clear that U + V = M: Let V be a supplement of U in M: Then K = V for every K V such that U +K = M: The modules every submodule of which has supplements are called supplemented modules. If every submodule of the R-module M has at least one supplement which is a direct summand in M; then M is called -supplemented. A submodule V of M is called a supplement in M if V is a supplement of a submodule in M: We say that a submodule U of the R-module M has ample supplements in M if, for every V M with U + V = M; there exists a supplement V 0 of U such that V 0

On certain properties of fuzzy group product operation

Dedekind's modular law plays a very important role in abstract group theory. We prove modular law for fuzzy groups and obtain conditions for a fuzzy subgroup to have a direct complement.

Distributive laws for commuting equivalence relations

In his paper (1942), Ore found necessary and sufficient conditions under which the modular and distributive laws hold in the lattice of equivalence relations on a set S. In the present paper, we consider commuting equivalence relations. It has been proved by Jónsson (1953) that the modular law holds in the lattice of commuting equivalence relations. We give some necessary and sufficient conditions for the distributive law and its dual to hold for commuting equivalence relations.

A Powerdomain of Possibility Measures

We provide a domain-theoretic framework for possibility theory by studying possibility measures on the lattice of opens &#x1D4AA;(X) of a topological space X. The powerspaces P[0,∞] (X) and P[0,1] (X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valuations by replacing ‘+’ with ‘V’ in their modular law. The functors above send continuous maps to sup-maps and continuous domains to completely distributive lattices; in the latter case they are locally continuous. Finite suprema of scalar multiples of point valuations form a basis of the powerdomains above if &#x1D4AA;(X) is the Scott-topology of a continuous domain. The notions of [0,1]- and [0,∞]-modules corresponds to that of continuous cones if addition on the reals and on the module is replaced by suprema. The powerdomain P[0,∞](D) is the free [0, ∞]-module over a continuous domain D.

Finitely based modular congruence varieties are distributive

R. Dedekind introduced the modular law, a lattice equation true in most of the lattices associated with classical algebraic systems, see [4]. Although this law is one of the most important tools for working with these lattices, it does not fully describe the equational properties of these lattices. This was made clear in [8] where B. Jonsson and the author showed that if any modular congruence variety actually satisfies the (stronger) Arguesian law. (A congruence variety is the variety generated by all the congruence lattices of the members of a variety of algebras.) In this note we show that no finite set of lattice equations is strong enough to describe the equational properties of the lattices associated with classical algebraic systems in the following strong sense: there is no modular, nondistributive congruence variety which has a finite basis for its equational theory. This question was posed by George McNulty in the problem session on lattice theory at the Jonsson symposium. Jonsson had asked a similar question in [17]. He asked, in Problem 9.12, whether there is a nontrivial variety whose congruence variety is neither the variety of all lattices nor the variety of all distributive lattices, but which is finitely based. Some partial results on this problem had been obtained previously. Let Vn denote the class of lattices embeddable into the lattice of subgroups of an abelian group of exponent n and let V0 be those lattices embeddable into the lattice of subspaces of a rational vector space. Thus HVn is the congruence variety associated with the variety of abelian groups of exponent n. In [7] it was shown that if L is a class of lattices contained in a modular congruence variety and containing Vn for some n not a prime, then L cannot be defined by finitely many first order axioms. It was also shown that every modular, nondistributive congruence variety contains HVn for some n. Mark Haiman, in his proof that there are Arguesian lattices which are not representable as lattices of permuting equivalence relations (solving an old problem of Jonsson), shows that HVp is not finitely based for any prime p, [11] and [12].

M n as a 0, 1-Sublattice of Con A Does not Force the Term Condition

For every n > 3 there exists a finite nonabelian algebra whose congruence lattice has Mn as a 0, 1-sublattice. This answers a question of R. McKenzie and D. Hobby. DEFINITION 0.1. Suppose L and L1 are bounded lattices. A copy of L (in L1) is any sublattice L' of L1 which is isomorphic to L. L' is a 0, 1-copy of L if it includes the least and greatest elements of L1; in this case L' is also called a 0, 1-sublattice of L1. DEFINITION 0. 2. For n > 1, Mn is the finite lattice of height 2 having exactly n atoms. For example, M6 is Suppose G is a group and N(G) is its lattice of normal subgroups. There is a trivial proof, using the commutator operation on normal subgroups, that if N(G) has a 0,1-copy of M3 then G is abelian. This same proof extends, via the general commutator theory of universal algebra, to any algebra A belonging to a variety whose congruence lattices satisfy the modular law. Here N(G) is replaced by Con A (the lattice of congruence relations of A), while 'abelian' means the following 'term condition': DEFINITION 0 . 3. An algebra A is abelian if for every n > 1, every (n + 1)-ary term t(x, Y1 ... Yn) in the language of A, and all a, b, c a .... acn dla ... adn E Al tA (a, c) = tA (a, d) iff tA(bC) = tA(b, d). In their forthcoming book on tame congruence theory [1], R. McKenzie and D. Hobby ask whether the above phenomenon Con A having M3 as a 0, 1-sublattice Received by the editors September 28, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 08A30; Secondary 06B99, 08A40.

Decidable varieties with modular congruence lattices

For a large collection of varieties we show that if the first-order theory of such a variety is decidable then the variety decomposes into the product of two well-known highly specialized varieties. For many varieties the decidability question then reduces to a decidability question about modules. A variety is a class of (abstract) algebras (belonging to some language) closed under the formation of direct products, subalgebras and homomorphic images. A variety 1/ is locally finite if every finitely-generated member of (/ is finite. A class of algebras K generates a variety 1/ if 1/ is the smallest variety containing K—then we say that \J is generated by K, written 1/ = V{K). A variety is finitely generated if it is generated by finitely many finite algebras, or equivalently by a single finite algebra. The kernel of a homomorphism is called a congruence, and the congruences of any algebra form a lattice. A variety is congruence modular, or we prefer to say just modular, if the lattice of congruences of every algebra in the variety satisfies the modular law. (Most of the well-studied varieties are modular; for example varieties of groups, rings, modules and lattices are modular. However,the variety of semigroups is not modular.) A variety \] is a product of two subvarieties | / j , |/2 ifl / j U l/2 generates (/ and there is a term b(x, y) such that M x \= b{x, y) = x, l/2 |= b(x, y) = y. If this is so we write 1/ = 1/ x <£) 1/ 2 , and then for every algebra A in 1/ there are (up to isomorphism) unique algebras Ax G (/p A2 G (/2 such that A = Ax x A2. A class K of first-order structures has a decidable theory if there is an effective procedure to determine precisely which first-order sentences are true of every member of K. A variety 1/ is a discriminator variety if it is generated by a set K for which there exists a ternary term t(xf y, z) such that K satisfies t(x, yy z) = x if x =£ y = z if x — y. In everyday mathematics such varieties appear only as highly specialized varieties of rings, or varieties associated with algebraic logics. Received by the editors December 29, 1980. 1980 Mathematics Subject Classification. Primary 03B25, 08B10, 08A05.

A Characterization of Identities Implying Congruence Modularity I

In his thesis and [24], J. B. Nation showed the existence of certain lattice identities, strictly weaker than the modular law, such that if all the congruence lattices of a variety of algebras satisfy one of these identities, then all the congruence lattices were even modular. Moreover Freese and Jónsson showed in [10] that from this “congruence modularity” of a variety of algebras one can even deduce the (stronger) Arguesian identity.These and similar results [3; 5; 9; 12; 18; 21] induced Jónsson in [17; 18] to introduce the following notions. For a variety of algebras , is the (congruence) variety of lattices generated by the class () of all congruence lattices θ(A), . Secondly if is a lattice identity, and Σ is a set of such, holds if for any variety implies .

Louise H. Chin and Alfred Tarski. Distributive and modular laws in the arithmetic of relation algebras. University of California publications in mathematics, n.s. vol. 1 no. 9 (1951), pp. 341–384.

Louise H. Chin and Alfred Tarski. Distributive and modular laws in the arithmetic of relation algebras. University of California publications in mathematics, n.s. vol. 1 no. 9 (1951), pp. 341–384.