J. D. Phillips

Diassociativity in Conjugacy Closed Loops

Abstract Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is power-associative and |Q| is finite and relatively prime to 6, then Q is a group. If Q is a finite non-associative extra loop, then 16 ∣ |Q|.

Every diassociative A-loop is Moufang

An A-loop is a loop in which every inner mapping is an automorphism. A problem which had been open since 1956 is settled by showing that every diassociative A-loop is Moufang.

C-LOOPS: EXTENSIONS AND CONSTRUCTIONS

C-loops are loops satisfying the identity x(y · yz) = (xy · 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 very transparent extensions; they can be built from small blocks arising from the underlying Steiner triple system. Using these extensions, we decide for which abelian groups K and Steiner loops Q there is a nonflexible C-loop C with center K such that C/K is isomorphic to Q. We discuss possible orders of associators in C-loops. Finally, we show that the loops of signed basis elements in the standard real Cayley–Dickson algebras are C-loops.

Finite Bruck loops

Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops X, showing that X is essentially the direct product of a Bruck loop of odd order with a 2-element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite 2-element Bruck loops are 2-loops, leaving open the question of whether such obstructions actually exist.

Commutants of Bol loops of odd order

1. IntroductionA loop (L,·) is a set L with a binary operation · : L×L → L such that(i) for given a,b ∈ L, the equations a· x = b and y · a = b have uniquesolutions x,y ∈ L, and (ii) there exists a neutral element 1 ∈ L satisfying1 · x = x · 1 = x for all x ∈ L. Basic references for loop theory are [1],[6]. We will use the usual juxtaposition conventions to avoid excessiveparenthesization, e.g., ab · c = (a· b) · c. The commutant of a loop L isthe setC(L) = {a ∈ L : ax = xa ∀x ∈ L}.The center of L is the set of all a ∈ C(L) such that a · xy = ax · y,x·ay = xa·y, and xy·a = x·ya for all x,y ∈ L. The center is a normalsubloop. For some varieties of loops, such as groups, the commutantand center coincide. For other varieties, the commutant is larger thanthe center, but is still “well-behaved” in the sense that it is a normalsubloop. However, the commutant of an arbitrary loop need not be asubloop at all, and even when it is, it need not be normal.The commutant is also known in the literature by other names suchas “Moufang center”, “commutative center”, or “centrum”. Since thisobject is not, in general, central in the sense of universal algebra, weprefer a term that does not suggest otherwise. Thus we have borrowedthe term “commutant”, which is used for a similar concept in other fields.A loop is said to be a (left) Bol loop if it satisfies the identity x(y·xz) =(x· yx)z for all x,y,z. A right Bol loop is similarly defined, and a loopwhich is both a left and right Bol loop is a Moufang loop. (This is one ofmany equivalent definitions; see [1], [6], [8].) In this paper, all Bol loopswillbeleftBolloops. The commutant ofaMoufangloopisasubloop, butit is an open problem to characterize precisely those Moufang loops forwhich the commutant is normal [2]. On the other hand, the commutant

When is the commutant of a Bol loop a subloop

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. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to 3, the commutant generates an abelian group of order dividing the order of the loop. This generalizes a well-known result for Moufang loops. After describing all extensions of a loop K such that K is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with a non-subloop commutant. In particular, we obtain all Bol loops of order 16 with a non-subloop commutant.

Automated theorem proving in quasigroup and loop theory

We survey all known results in the area of quasigroup and loop theory to have been obtained with the assistance of automated theorem provers. We provide both informal and formal descriptions of selected problems, and compare the performance of selected state-of-the art first order theorem provers on them. Our analysis yields some surprising results, e.g., the theorem prover most often used by loop theorists does not necessarily yield the best performance.

Quasiprimitivity and quasigroups

It is well known that Q is a simple quasigroup if and only if Mlt Q acts primitively on Q . Here we show that Q is a simple quasigroup if and only if Mlt Q acts quasiprimitively on Q , and that Q is a simple right quasigroup if and only if RMlt Q acts quasiprimitively on Q .

Linear groupoids and the associated wreath products

A groupoid identity is said to be linear of length 2k if the same k variables appear on both sides of the identity exactly once. We classify and count all varieties of groupoids defined by a single linear identity. For k=3, there are 14 nontrivial varieties and they are in the most general position with respect to inclusion. Hentzel et al. [Hentzel, I.R., Jacobs, D.P., Muddana, S.V., 1993. Experimenting with the identity (xy)z=y(zx). J. Symbolic Comput. 16, 289-293] showed that the linear identity (xy)z=y(zx) implies commutativity and associativity in all products of at least five factors. We complete their project by showing that no other linear identity of any length behaves this way, and by showing how the identity (xy)z=y(zx) affects products of fewer than five factors; we include distinguishing examples produced by the finite model builder Mace4. The corresponding combinatorial results for labelled binary trees are given. We associate a certain wreath product with any linear identity. Questions about linear groupoids can therefore be transferred to groups and attacked by group-theoretical computational tools, e.g., GAP. Systematic notation and diagrams for linear identities are devised. A short equational basis for Boolean algebras involving the identity (xy)z=y(zx) is presented, together with a proof produced by the automated theorem prover OTTER.

Bruck Loops with Abelian Inner Mapping Groups

Bruck loops with abelian inner mapping groups are centrally nilpotent of class at most 2.