Inverse semigroups with idempotent-fixing automorphisms

A celebrated result of J. Thompson says that if a finite group $G$ has a fixed-point-free automorphism of prime order, then $G$ is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.Comment: 7 pages in ijmart styl

An elegant 3-basis for inverse semigroups

It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following three identities: [\quad x=(xx')x \qquad \quad (xx')(y'y)=(y'y)(xx') \qquad \quad (xy)z=x(yz"). ] The goal of this note is to prove the converse, that is, we prove that an algebra of type $$ satisfying these three identities is an inverse semigroup and the unary operation coincides with the usual inversion on such semigroups.Comment: 4 pages; v.2: fixed abstract; v.3: final version with minor changes suggested by referee, to appear in Semigroup Foru

Minimal paths in the commuting graphs of semigroups

Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted \cg(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 $a\ne b$ 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 $|X|\geq4$, then, with a few exceptions, the diameter of \cg(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 \cg(S) is $n$. We also study the left paths in \cg(S), that is, paths $a_1-a_2-...-a_m$ such that $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for all i\in \{1,\ldot, m\}. We prove that for every positive integer $n\geq2$, 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.Comment: 23 pages; v.2: Lemma 2.1 corrected; v.3: final version to appear in European J. of Combinatoric

Estimating the Stochastic Discount Factor without a Utility Function

In this paper we take seriously the consequences of the Pricing Equation in constructing a novel consistent estimator of the stochastic discount factor (SDF) using panel data. Under general conditions it depends exclusively on appropriate averages of asset returns, and its computation is a direct exercise, as long as one has enough observations to fit our asymptotic results. We identify the logarithm of the SDF using the fact that it is the serial correlation "common feature" in every asset return of the economy. Our estimator does not depend on any parametric function representing preferences, or on consumption data. This property allows its use in testing different preference specifications commonly employed in finance and in macroeconomics, as well as investigating the existence of several puzzles involving intertemporal substitution, such as the equity-premium puzzle. It is also straightforward to construct an estimator of the risk-free rate based on our SDF estimator. When applied to quarterly data of U.S.$ real returns from 1972:1 through 2002:4, our estimator of the SDF is close to unity most of the time and yields an equivalent average annual real discount rate of 2.46%. When we examined the appropriateness of different functional forms to represent preferences, we concluded that standard preference representations used in the literature on intertemporal substitution cannot be rejected by the data. Moreover, estimates of the relative risk-aversion coefficient are close to what can be expected a priori -- between 1 and 2, statistically significant, and not different than unity in testing. A direct test of the equity-premium puzzle using our SDF estimator cannot reject the null that the discounted equity premium in the U.S. has mean zero. However, when consumption-based SDF estimates are employed in the same test, the null is rejected. Further empirical investigation shows that our SDF estimator has a large negative correlation with the equity premium, whereas that of consumption-based estimates are usually too small in absolute value, generating the equity-premium puzzlecommon features, stochastic discount factor

Embedding properties of endomorphism semigroups

Denote by PSelf(X) (resp., Self(X)) the partial (resp., full) transformation monoid over a set X, and by Sub(V) (resp., End(V)) the collection of all subspaces (resp., endomorphisms) of a vector space V. We prove various results that imply the following: (1) If X has at least two elements, then Self(X) has a semigroup embedding into the dual of Self(Y) iff card(Y) >= 2^card(X). In particular, if X has at least two elements, then there exists no semigroup embedding from Self(X) into the dual of PSelf(X). (2) If V is infinite-dimensional, then there are no embedding from (Sub(V),+) into (Sub(V),\cap) and no semigroup embedding from End(V) into its dual. (3) Let F be an algebra freely generated by an infinite subset X. If F has less than 2^card(X) operations, then End(F) has no semigroup embedding into its dual. The cardinality bound 2^card(X) is optimal. (4) Let F be a free left module over a left aleph one - noetherian ring (i.e., a ring without strictly increasing chains, of length aleph one, of left ideals). Then End(F) has no semigroup embedding into its dual. (1) and (2) above solve questions proposed by B. M. Schein and G. M. Bergman. We also formalize our results in the settings of algebras endowed with a notion of independence (in particular independence algebras).Comment: To appear in Fundamenta Mathematica