ON A NEW APPROACH TO THE DUAL SYMMETRIC INVERSE MONOID $\mathcal{I}_{X}^{\ast}$

We construct the inverse partition semigroup[Formula: see text], isomorphic to the dual symmetric inverse monoid[Formula: see text], introduced in [6]. We give a convenient geometric illustration for elements of [Formula: see text]. We describe all maximal subsemigroups of [Formula: see text] and find a generating set for [Formula: see text] when X is finite. We prove that all the automorphisms of [Formula: see text] are inner. We show how to embed the symmetric inverse semigroup into the inverse partition one. For finite sets X, we establish that, up to equivalence, there is a unique faithful effective transitive representation of [Formula: see text], namely to [Formula: see text]. Finally, we construct an interesting [Formula: see text]-cross-section of [Formula: see text], which is reminiscent of [Formula: see text], the [Formula: see text]-cross-section of [Formula: see text], constructed in [4].

COVERINGS OF LINKS AND A GENERALIZATION OF RILEY’S CONJECTURE B

Let k be a knot in S3, ψ: π1(S3–k)→Zα1 * Zα2 an epimorphism sending a meridian to an element of length 2 and ω: Zα1 * Zα2→Sσ a transitive representation into a symmetric group. Information is given (in Theorem B##) on the homologies of the unbranched and branched covers associated to ωψ which generalizes a conjecture of Riley. When k is a torus knot these homologies are actually computed.

Clebsch–Gordan decomposition for transitive representations

The method of evaluation of Clebsch–Gordan coefficients to calculate the direct product of transitive representations for a finite group is developed. This procedure is based on the Mackey theorem and on the canonical realization of transitive representations. It is shown that the Clebsch–Gordan decomposition is associated with a natural interpretation of an orbit of a resultant transitive representation and it is analogous to a fibre bundle with the fibration being determined by constituent transitive representations.

‘On projective planes of type (6, m)’

Alan Rahilly has pointed out to me that the proof of the Theorem in my paper (2) is incomplete. This correction will now complete it. I would also like to acknowledge here that the results which are actually established in (2) can also be found in the paper (5) by Praeger and Rahilly.The trouble was in the proof of Proposition 2. Although the group G has a subgroup H1 which intersects each of its conjugates trivially, the same is not necessarily true of the image H1N/N of H1 in the 2-transitive representation G/N of G referred to in the paper. A theorem of M. E. O'Nan from (4) was used along with Proposition 2 to establish my theorem. What should have been done was to look at O'Nan's results more deeply and combine them with other results. Here is the way that it is done.There is nothing actually wrong with the argument in my paper; it is just that it is not complete. The proof here will continue the argument of (2) using the notations established there.

A Note on Groups of Ree Type

The nonsolvable R-groups as defined by Walter [3] are groups of orders (q3+l)q3(q — 1), q = 32n+1, n ≥ 0. These are the groups of Ree type discussed by Ward [4] together with the Ree group R(3) of order 28.27.2. The R-group with parameter q has a doubly transitive representation of degree q3+1 but in this note we will prove that it cannot contain a sharply doubly transitive subset.