Related papers: On The Universality Of Central Loops
The right(left) derivative, $a^{-1},e-$ and $e,a^{-1}-$ isotopes of a C-loop are shown to be C-loops. Furthermore, for a central loop $(L,F)$, it is shown that $\big\{F,F^{a^{-1}},F_{a^{-1},e}\big\}$ and…
The representation sets of central loops are investigated and the results obtained are used to construct a finite C-loop. It is shown that for certain types of isotopisms, the central identities are isotopic invariant.
C-loops are loops satisfying $x(y(yz))=((xy)y)z$. They often behave analogously to Moufang loops and they are closely related to Steiner triple systems and combinatorics. We initiate the study of C-loops by proving: (i) Steiner loops are…
We study loops which are universal (that is, isotopically invariant) with respect to the property of flexibility ($xy\cdot x = x\cdot yx$). We also weaken this to semi-universality, that is, loops in which every left and right isotope is…
C-loops are loops satisfying the identity $x(y\cdot yz) = (xy\cdot y)z$. We develop the theory of extensions of C-loops, and characterize all nuclear extensions provided the nucleus is an abelian group. C-loops with central squares have…
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 notion of the holomorph of a generalized Bol loop (GBL) is characterized afresh. The holomorph of a right inverse property loop (RIPL) is shown to be a GBL if and only if the loop is a GBL and some bijections of the loop are right…
A question associated with the 2005 open problem of Michael Kinyon (Is every Osborn loop universal?), is answered. Two nice identities that characterize universal (left and right universal) Osborn loops are established. Numerous new…
A Smarandache quasigroup(loop) is shown to be universal if all its f,g-principal isotopes are Smarandache f,g-principal isotopes. Also, weak Smarandache loops of Bol-Moufang type such as Smarandache: left(right) Bol, Moufang and extra loops…
An algebraic process for the construction of an autotopism for a non-Steiner C-loop is described and this is demonstrated with an example using a known finite C-loop. In every C-loop, two of its parastrophes are not equivalent(equal) it, if…
A loop $(X,\circ)$ is said to be a Bruck loop if it satisfies the (right) Bol identity $((z\circ x)\circ y)\circ x = z\circ ((x\circ y)\circ x)$ and the automorphic inverse property $(x\circ y)^{-1}=x^{-1}\circ y^{-1}$. If $X$ is a finite…
The circumcircle of a planar convex polygon P is a circle C that passes through all vertices of P. If such a C exists, then P is said to be cyclic. Fix C to have unit radius. While any two angles of a uniform cyclic triangle are negatively…
Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. A half-isomorphism $f : G \longrightarrow K$ between multiplicative systems $G$ and $K$ is a…
A loop is shown to be a universal Osborn loop if and only if it has a particular simplicial complex. A loop is shown to be a universal Osborn loop and obeys two new identities if and only if it has another particular simplicial complex. A…
If two loops are isomorphic, then it is shown that their holomorphs are also isomorphic. Conversely, it is shown that if their holomorphs are isomorphic, then the loops are isotopic. It is shown that a loop is a Smarandache loop if and only…
This paper defines new intersection homology groups. The basic idea is this. Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles. A local-global cycle is defined to be a family of such…
A left Bol loop is a loop satisfying $x(y(xz)) = (x(yx))z$. The commutant of a loop is the set of elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order $2k$, $k$ odd, the commutant is a subloop.…
An \emph{automorphic loop} (or \emph{A-loop}) is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and $(xy)^{-1} = x^{-1}y^{-1}$ holds. Let $Q$ be a finite commutative A-loop and $p$ a…
The notion of the holomorph of a generalized Bol loop and generalized flexible-Bol loop are characterized. With the aid of two self-mappings on the holomorph of a loop, it is shown that: the loop is a generalized Bol loop if and only if its…
A ring $R$ with center $C$ is said to be centrally essential if the module $R_C$ is an essential extension of the module $C_C$. In this paper, we study properties of ideals of centrally essential rings, centrally essential quaternion…