In this paper, we deal with Molaei’s generalized groups. We definethe notion of a fuzzy generalized subgroup with respect to a t-norm (orT-fuzzy generalized subgroup) and give some related properties. Especially,we state and prove the Representation Theorem for these fuzzy generalizedsubgroups. Next, using the concept of continuity of t-norms we obtain a correspondencebetween TF(G), the set of ...

We prove that a group G is Abelian whenever (1) it is nilpotent and the lattice of normal subgroups of G is isomorphic to the subgroup lattice of an Abelian group or (2) there exists a non-torsion Abelian group B such that the normal subgroup lattice of B × G is isomorphic to the subgroup lattice of an Abelian group. Using (2), it is proved that an Abelian group A can be determined in the class...

Nakamura [N] introduced the G-Hilbert scheme G-Hilb C3 for a finite subgroup G ⊂ SL(3, C), and conjectured that it is a crepant resolution of the quotient C3/G. He proved this for a diagonal Abelian group A by introducing an explicit algorithm that calculates A-HilbC3. This note calculates A-Hilb C3 much more simply, in terms of fun with continued fractions plus regular tesselations by equilate...

the present note arises from the author's talk at the conference ``ischia group theory 2014''. for subgroups $fle n$ of a group $g$ denote by $lat(f,n)$ the set of all subgroups of $n$, containing $f$. let $d$ be a subgroup of $g$. in this note we study the lattice $ll=lat(d,g)$ and the lattice $ll'$ of subgroups of $g$, normalized by $d$. we say that $ll$ satisfies sandwich classification theo...

We prove that whenever a Kac-Moody group over a finite field is a lattice of its buildings, it has a fundamental domain with respect to which the induction cocycle is L for any p∈ [1; +∞). The proof uses elementary counting arguments for root group actions on buildings. The applications are the possibility to apply some lattice superrigidity, and the normal subgroup property for Kac-Moody latti...

 ‎Let $tilde{F}=(Q,S,tilde{R},Z,omega,tilde{delta},‎ F_1,F_2)$‎ ‎be a general fuzzy automaton and the set of its states be a group.‎ ‎The aim of this paper is the study of applications of a group in a‎ ‎general fuzzy automaton‎. ‎For this purpose‎, ‎we define the concepts‎ ‎of fuzzy normal kernel of a general fuzzy automaton‎, ‎fuzzy kernel‎ ‎of a general fuzzy automaton‎, ‎adjustable‎, ‎multip...

in this paper, we deal with molaei’s generalized groups. we definethe notion of a fuzzy generalized subgroup with respect to a t-norm (ort-fuzzy generalized subgroup) and give some related properties. especially,we state and prove the representation theorem for these fuzzy generalizedsubgroups. next, using the concept of continuity of t-norms we obtain a correspondencebetween tf(g), the set of ...

Let L be a bounded lattice, let [a, b] and [c, d] be intervals of L, and let φ : [a, b] → [c, d] be an isomorphism between these two intervals. Let us consider the algebra L↔ φ = 〈L;∧,∨, φ, φ−1〉, which is a lattice with two partial unary operations. We construct a bounded lattice K (in fact, a convex extension of L) such that the congruence lattice of L↔ φ is isomorphic to the congruence lattic...

By a new partial ordering relation “ ≤ ” the set of convex sublattices CS(L) of a lattice L is again a lattice. In this paper we establish some results on the pseudocomplementation of (CS(L); ≤). We show that a lattice L with 0 is dense if and only if CS(L) is dense. Then we prove that a finite distributive lattice is a Stone lattice if and only if CS(L) is Stone. We also prove that an upper co...

The paper is a supplement to [2]. Let L be a lattice and U an o-symmetric convex body inR n. The Minkowski functional I[ I1~ of U, the polar body U ~ the dual lattice L*, the covering radius/z(L, U), and the successive minima ~.i(L, U), i = 1 . . . . . n, are defined in the usual way. Let 12, be the family of all lattices in R". Given a convex body U, we define mh(U) = sup max ~.i(L, U)~.n-i+l(...