Abstract. We show that the skew-symmetrized product on every Leibniz algebra E can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on E which, when E is a Lie algebra, is derived from the integrated adjoint representation. We apply this construction to realize the bracket operations on the sections of Courant algebroids and on the "omni-Lie algebras" recently introduced by the second author.

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Let S be a finite non-commutative semigroup. The commuting graph of S, denoted G(S), is the graph whose vertices are the non-central elements of S and whose edges are the sets {a, b} of vertices such that a = b and ab = ba. Denote by T (X) the semigroup of full transformations on a finite set X. Let J be any ideal of T (X) such that J is different from the ideal of constant transformations on X. We prove that if |X| ≥ 4, then, with a few exceptions, the diameter of G(J) is 5. On the other hand, we prove that for every positive integer n, there exists a semigroup S such that the diameter of G(S) is n.We also study the left paths in G(S), that is, paths a1 − a2 − · · · − am such that a1 = am and a1ai = amai for all i ∈ {1, . . . , m}. We prove that for every positive integer n ≥ 2, except n = 3, there exists a semigroup whose shortest left path has length n. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein.2010 Mathematics Subject Classification. 05C25, 05C12, 20M20.Let S be a semigroup. We denote by G E (S) the subgraph of G(S) induced by the non-central idempotents of S. The graph G E (S) is said to be the idempotent commuting graph of S. We first determine the diameter of G E (J r ). This approach is justified by the following lemma.Lemma 2.1. Let 2 ≤ r < n and let a, b ∈ J r be such that ab = ba. Suppose a−a 1 −a 2 −· · ·−a k −b (k ≥ 1) is a minimal path in G(J r ) from a to b. Then there are idempotents e 1 , e 2 , . . . , e k ∈ J r such that a − e 1 − e 2 − · · · − e k − b is a minimal path in G(J r ) from a to b.Proof. Since J r is finite, there is an integer p ≥ 1 such that e 1 = a p 1 is an idempotent in J r . Note that e 1 / ∈ Z(J r ) since for any x ∈ X − im(e 1 ), e 1 does not commute with c x ∈ J r , where c x is the constant transformation with im(c x ) = {x}. Since a 1 commutes with a and a 2 , the idempotent e 1 = a p 1 also commutes with a and a 2 , and so a − e 1 − a 2 − · · · − a k − b. Repeating the foregoing argument for a 2 , . . . , a k , we obtain idempotents e 2 , . . . , e k in J r such that a − e 1 − e 2 − · · ·− e k − b. Since the path a − a 1 − a 2 − · · · − a k − b is minimal, it follows that a, e 1 , e 2 , . . . , e k , b are pairwise distinct and the path a − e 1 − e 2 − · · · − e k − b is minimal.

Abstract. An automorphic loop (or 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 prime. The loop Q has order a power of p if and only if every element of Q has order a power of p. The loop Q decomposes as a direct product of a loop of odd order and a loop of order a power of 2. If Q is of odd order, it is solvable. If A is a subloop of Q, then |A| divides |Q|. If p divides |Q|, then Q contains an element of order p. If there is a finite simple nonassociative commutative A-loop, it is of exponent 2.

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|. * Author supported by NSF Grant DMS-0097881 2 Proof. RCC and LCC assert that L y L x = L x L f (x,y) and R y R x = R x R g(x,y) .Thus, in a CC-loop, the left multiplications are closed under conjugation and the right multiplications are closed under conjugation; hence the name "conjugacy-closed".These loops have a number of interesting properties, surveyed in Sections 2 and 3; for example, by [10], the left and right inner mappings are automorphisms. These properties allow a rather detailed structural analysis to be made; in particular, all CC-loops of orders p 2 and 2p (for primes p) are known (see [13]). This paper yields additional structural information about CC-loopsespecially for the ones which are power-associative (that is, each x is a group) or diassociative (that is, each x, y is a group).It is shown in [11] that the CC-loops which are diassociative (equivalently, Moufang) are the extra loops studied by Fenyves [8,9]. By [9], if Q is an extra loop, then Q/N(Q) is a boolean group (where N(Q) is the nucleus). It is immediate that a finite extra loop of odd order is a group. We show here (Corollary 7.7) that a finite power-associative CC-loop of order relatively prime to 6 is a group. The "6" cannot be improved, since there are non-associative power-associative CC-loops of order 16 (e.g., the Cayley loop) and of order 27 (see Section 9) (we do not know if there are ones of order divisible by 6 but not by 4 or 9). Also, one cannot drop the "power-associative", since by [10], there are non-power-associative CC-loops of order p 2 for every odd prime p.More generally, we show that every power-associative CC-loop satisfies a weakening of diassociativity -namely, x, y is a group whenever x is a cube and y is a square. Then, if |Q| is relatively prime to 6, every element must be a sixth power by the Lagrange property, so that Q is diassociative, and hence an extra loop of odd order, and hence a group. Of course, we must verify that the Lagrange property really holds for CC-loops, since it can fail for loops in general. This is easy to do (see Corollary 3.2) using the result of Basarab [2]. He showed that for any CC-loop, Q/N(Q) is an abelian group (this answers a question from [10]); we include a proof of this here (see Theorem 3.1), since it is fairly short using the notion of autotopy (see Belousov [3] II §3 or Bruck [5] VII §2), together with some facts about the autotopies of CC-loops proved by Goodaire and Robinson (see [10] and Section 2).We also establish two theorems about general CC-loops. First, whenever S ⊆ Q and S associates in the sense that x · yz = xy · z holds for all x, y, z ∈ S, we prove that S also associates, and hence is a group (see Cor...

A loop whose inner mappings are automorphisms is an automorphic loop (or A-loop). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate the classification of commutative A-loops of small orders and also of order p 3 , where p is a prime.

We solve a problem of Belousov which has been open since 1967: to characterize the loop isotopes of F-quasigroups. We show that every F-quasigroup has a Moufang loop isotope which is a central product of its nucleus and Moufang center. We then use the loop to reveal the structure of the associated F-quasigroup.

Abstract. 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 of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely 2-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop Q of odd order is solvable, contains an element of order p for every prime p dividing |Q|, and |S| divides |Q| for every subloop S of Q.There are no finite simple nonassociative commutative automorphic loops, and there are no finite simple nonassociative automorphic loops of order less than 2500. We show that if Q is a finite simple nonassociative automorphic loop then the socle of the multiplication group of Q is not regular. The existence of a finite simple nonassociative automorphic loop remains open.Let p be an odd prime. Automorphic loops of order p or p 2 are groups, but there exist nonassociative automorphic loops of order p 3 , some with trivial nucleus (center) and of exponent p. We construct nonassociative "dihedral" automorphic loops of order 2n for every n > 2, and show that there are precisely p − 2 nonassociative automorphic loops of order 2p, all of them dihedral.