Related papers: Smarandache Loops

Although little can be gleaned about a loop with the property that its squares are, say, left nuclear ($xx\cdot yz = (xx\cdot y)z$), if its squares are also, say, middle nuclear ($(x\cdot yy)z = x(yy\cdot z)$), then the loop exhibits more…

The NAFIL is a finite loop in which every element has a unique (two-sided)inverse. NAFIL loops can be classified into two types: composite (with at least one non-trivial subsystem) and non-composite or plain (without any non-trivial…

The isotopic invariance or universality of types and varieties of quasigroups and loops described by one or more equivalent identities has been of interest to researchers in loop theory in the recent past. A variety of quasigroups(loops)…

The paper focuses on the structure of fundamental sequences of ordinals smaller than $\epsilon_0$. A first result is the construction of a monadic second-order formula identifying a given structure, whereas such a formula cannot exist for…

A binary operation on any set induces a binary operation on its subsets. We explore families of subsets of a group that become a group under the induced operation and refer to such families as power groups of the given group. Our results…

We propose a way of associating to each finitely generated monoid or semigroup a formal language, called its loop problem. In the case of a group, the loop problem is essentially the same as the word problem in the sense of combinatorial…

For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to \Aut \,(G, \cdot),~~a \mapsto \lambda_a,$ where $\lambda_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can also be…

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops. We study uniquely 2-divisible automorphic loops, particularly automorphic loops…

In program semantics and verification, reasoning about loops is complicated by the need to produce two separate mathematical arguments: an invariant, for functional properties (ignoring termination); and a variant, for termination (ignoring…

A saddle loop is a germ of a holomorphic foliation near a homoclinic saddle connection. We prove that they are classied by their Poincar{\'e} rst-return map. We also prove that they are formally rigid when the Poincar{\'e} map is…

We investigate the problem of defining group or loop structures on spheres, where by ''sphere'' we mean the level set q(x) = c of a general K-valued quadratic form q, for an invertible scalar c. When K is a field and q non-degenerate, then…

Given a real, symmetric matrix S, we define the slice through S as being the connected component containing S of two orbits under conjugation: the first by the orthogonal group, and the second by the upper triangular group. We describe some…

Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a lambda-ring. However, a priori the images of S under the lambda-operations can only be computed implicitly. In this paper we establish an…

We construct a Moufang loop $M$ of order $3^{19}$ and a pair $a,b$ of its elements such that the set of all elements of $M$ that associate with $a$ and $b$ does not form a subloop. This is also an example of a nonassociative Moufang loop…

We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects…

An arrangement of pseudocircles is a finite set of oriented closed Jordan curves each two of which cross each other in exactly two points. To describe the combinatorial structure of arrangements on closed orientable surfaces, in (Linhart,…

We study the combinatorial and structural properties of the circle map sequences. We introduce an embedding procedure which gives a map from the hull(closure of the set of translates) to the sequence of embedding operations through which we…

In general the endomorphisms of a non-abelian group do not form a ring under the operations of addition and composition of functions. Several papers have dealt with the ring of functions defined on a group which are endomorphisms when…

We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite dimensional manifold. We succeed for a particular type of piecewise-smooth loops. We also examine the…

In this paper a variety of issues are discussed, Schur ring, $S$-sets, circulant orbits, decimation operator and Hadamard matrices and their relation between them is shown. Firstly we define the complete $S$-sets. Next, we study the…