A homomorphism theorem for projective planes

Chapter 4 - Projective Planes

André (1) gave a construction for translation planes from abelian groups possessing “congruences” of subgroups. Schwerdtfeger (3) constructed the plane over a field F from the group of substitutions x → ax + b (a, b ∊ F; a ≠ 0). In this note we describe a construction (inspired by Schwerdtfeger's work), from groups, of planes which are duals of near-field planes.If a plane is (l, m)-transitive (cf. 2, p. 67) for some pair of distinct lines l, m, then the central collineations ϕ with axis m and centre on I may be identified with the “proper” points (that is, points not on I or m) of the plane once an origin O is chosen (not on l or m):

The document outlined the definition of F-pseudo Z-b-subalgebra in addition to introducing fundamental properties. Multiple theorems have been established throughout the discussion using appropriate examples. We introduced a new concept after presenting the material.: (fuzzy set μdz fj) of pseudo Z-algebra. This research examines both the homomorphic images and inverse images from this study. We denote b-sg to represent b-subalgebra in this article.

In this paper the notion of fuzzy BE-algebras are introduced. We state and prove some theorems in fuzzy BE-algebras and level subalgebras. Some characterizations of fuzzy subalgebras of BE-algebras are given. Using fuzzy subalgebras, characterizations of Artinian and Noetherian BE-algebras are also established. Finally the notion of fuzzy topological BE-algebras is introduced. We state and prove the Foster’s results on homomorphic images and inverse images in fuzzy topological BE-algebras.

In this article, we apply the concept of a t -intuitionistic fuzzy set to PMS-ideals in PMS-algebras. The notion of the t -intuitionistic fuzzy PMS-ideal of PMS-algebra is introduced, and several related properties are studied. The relationships between a t -intuitionistic fuzzy PMS-ideal and a t -intuitionistic fuzzy PMS-subalgebra of a PMS-algebra, as well as the relationships between an intuitionistic fuzzy PMS-ideal and a t -intuitionistic fuzzy PMS-ideal are discussed in detail. A condition for an intuitionistic fuzzy set to be a t -intuitionistic fuzzy PMS-ideal is provided. The t -intuitionistic fuzzy PMS-ideals of PMS-algebra are described using their α , β level cuts. The homomorphism of a t -intuitionistic fuzzy PMS-ideal of a PMS-algebra is studied, and its homomorphic image and inverse image are explored. The Cartesian product of any two t -intuitionistic fuzzy PMS-ideals is discussed, and some related results are derived. The Cartesian product of the t -intuitionistic fuzzy PMS-ideals is also characterized using its α , β level cuts. The strongest t -intuitionistic fuzzy PMS-relation in a PMS-algebra is defined. Finally, the relationships between the strongest t -intuitionistic fuzzy PMS-relation and t -intuitionistic fuzzy PMS-ideal are studied.

In this paper, the notion of cubic subalgebras of BCH-algebras are introduced. Some characterization of cubic subalgebras of BCH-algebras are given. The homomorphic image and inverse image of cubic subalgebras are studied and investigated some related properties

The aim of this paper is to extend fuzzifying BCH algebra by introducing a unary predicate and by adopting the semantic method of continuous lattice valued logic. We define the concepts of fuzzifying sub algebras, fuzzifying ideals, fuzzifying closed ideals, fuzzifying q-ideals and fuzzifying implicative ideals in BCH-algebras, and discuss the relations among them. How to deal with the homomorphic images and inverse images of fuzzifying ideals, fuzzifying closed ideals, fuzzifying q-ideals and fuzzifying implicative ideals are studied. The product of two fuzzifying ideals (fuzzifying closed ideals, fuzzifying q-ideals and fuzzifying implicative ideals) is investigated.

In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. MSC: 03G25, 06D30.

A probabilistic automaton (pa) is a fuzzy automaton, in which, sum of all fuzzy memberships of the initial states is one and from each state, on an input symbol, the sum of the fuzzy memberships of the transitions is one. In pa, it is shown that, if h : Γ* → Σ* is a morphism and h–1 (λ) = λ, then the image of a recognisable subset of Γ* is a recognisable subset of Σ* and if h is fine, then the inverse image of recognisable subset of Σ* is recognisable. It is also proved that the shuffle product of any two recognisable sets is recognisable.

The concept and properties of cubic sets were first introduced and investigated by Jun et al. in 2011. In this paper, we first introduce the concept of cubic sets in UP-subalgebras and consider the UP-ideals of a UP-algebra. Relationships between the cubic UP-subalgebras and the cubic UP-ideals of a UP-algebra are investigated. In addition, the homomorphic images and the inverse images of the cubic UP-subalgebras and the cubic UP-ideals are also studied. Related properties of UP -algebras and their UP-ideals are explored and discussed. Finally, the Cartesian product of cubic UP-algebras are investigated.

In this paper, the notions of hesitant fuzzy dot subalgebras, hesitant fuzzy normal dot subalgebras, and hesitant fuzzy dot ideals of B-algebras are presented, and some of their features are examined, in this study. The homomorphic image and inverse image of hesitant fuzzy dot subalgebras, as well as the hesitant fuzzy dot ideal, are investigated. We also explore some related characteristics of hesitant fuzzy relations on the family of hesitant fuzzy dot subalgebras and hesitant fuzzy dot ideal of B-algebras.

In this paper, the notion of interval-valued fuzzy SA-ideals (briefly i-v fuzzy SA-ideal) in SA-algebras is introduced. Several theorems are stated and proved. The image and inverse image of i-v fuzzy SA-ideals are defined and how the homomorphic images and inverse images of i-v fuzzy SA-ideals become i-v fuzzy SA-ideals in SA-algebras is studied as well.

In this paper, we introduce concepts of T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> -interval- valued fuzzy left ideal, Th-interval-valued fuzzy right ideal, T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> -interval-valued fuzzy ideal and extension principle of interval-valued fuzzy set, investigate some of their properties and structural characteristics and propose some theorems for homomorphic image and its inverse image on T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> -interval-valued fuzzy ideal.

Two symmetric presentations for the Rudvalis sporadic simple group Ru are given, and corresponding presentations in terms of generators and relations are deduced. The first considers the linear group L4(2) acting imprimitively on 105 letters and so analyses a progenitor of form 2*105 : L4(2). Certain short relations follow from the basic lemmas of symmetric generation of groups and the resulting homomorphic image is shown to be Ru. In the second approach we work within Ru to show that the group is generated (uniquely up to conjugation) by a set of seven involutions whose set normalizer in Ru is the linear group L3(2), and which is such that any set of four of these involutions, no three of which lie on a line of the underlying projective plane, generates a copy of the Tits simple group 2F4(2)′. Thus Ru is obtained as an image of the progenitor 2*7 : L3(2).

We study surface knots in 4–space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the total width) of a surface knot is a numerical invariant related to the number of points in the inverse image of a point in each of the regions. We determine the widths of certain surface knots and characterize those surface knots with small total widths. Relation to the surface braid index is also studied.