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availability of suitable refuges (e.g. natural or artificial) is likely to be a central component for the conservation of many reptile species. The combination of empirical and experimental data further demonstrates that great attention must be paid to the structure and distribution of the refuges and that simple practical actions can effectively improve habitat quality for threatened species.

Dedicated to Professor A. H. Clifford for his 65th birthdayi° Let A be a semigroup, and G be a group-valued functor on the set A preordered by left divisibility; G assigns to each a~A a group G and to each pair (g,~)~ G~ • The main conwith ~ ¢ aA 1 a homomorphlsm ~ : G cept in this paper is that of an extension of G by A :that is a semigroup S together with a congruence C such that S/C ---A , and, for each ecA, a simply transitive group action of G a on the corresponding C-class C, such that q0aa~g.ab = (g.a)b holds in S whenever gcG~, a~C , bgC~oThe main results are as follows. Given any semlgroup S, certain congruences on S, which we call left coset congruences, give rise to a description of S as an extension . This is an arbitrary selection of significant results; the reader will find a bibliography in [16] and topological examples in [2]° In each of these examples the general theory might not provide the best proof, but it explains why a construction is possible, and gives a way to find such a construction as well as the general form of the multiplication. The general theory has also begun to yield new results; in particular, a construction of all finite commutative semlgroups [13] (partly by induction on S/~ ) which also applies to the finitely generated com- Of course all the important applications of the theory so far concern regular or commutative semigroups, for which this case is vacuouso Future applications, however, may not 202GRILLET be so restricted; it may then be advantageous to build from congruences that are as large as possible. The added sharpness is clear, at a more basic level, on the group which a left coset congruence attaches to a S-class (much as does): this group need not be contained in the Sch~tzenber-ger group (it may even be greater) and so gives a different kind of structural insight.3. Our first basic idea is that the usual Sch~tzenber-get group construction may be carried out with subsets which need not be Z-classes nor even be contained in Z-classes. We call them left cosets and study them in section I, The basic algebraic fact here is that the "right hand side" maps X can be recovered from purely "left hand side" concepts, which yields a sort of a posteriori duality.In section 3 we also point out that the X maps need not be homomorphisms. This phenomenon (bad behavior) can happen in left coset extensions and thus cannot be ignored.In section 4 we show that in a left coset extension bad We now select the following disjoint union of the O a ::. Formula (3.4) says that we can define a multiplication on T by : gives the action of G on S ; associativity in G insures that it is a group action, and clearly it is simply transitive. Finally, we show that this is indeed an extension;that it, g.xy = (g.x)y for all g¢ G, x,y¢S. Observe that (due to our notation) the left action of g has, formally, the same effect on x as left multiplication by g in G,while right multiplication by y (in S ) has, formally, the same effect as right multiplication in G by either I, a or b ; therefore the...

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The tensor product of semigroups is defined like the tensor product of modules, by means of multilinear mappings. Surprisingly enough, it has most of the important properties of its homological cousin, together with some others of its own, so that, in view of such results as [4], it is not unreasonable to hope that it will become a very helpful tool for the study of semigroups. The only thing it lacks as an operation is associativity; this is apparently due to the use of noncommutative semigroups throughout this paper, and results in our concentrating on the tensor product of two semigroups, even though such restriction is not necessary for some of our results.In §1 we give the definition and some examples: If B is a one-element semigroup, then A ® Bis the largest idempotent homomorphic image of A ; if B is an infinite cyclic semigroup, then A ® Bis the largest normal homomorphic image N(A) of A (normal means that (xy)n = xnyn holds identically for all n). In §2 we prove the existence of the tensor product of any family of semigroups. §3 brings a fundamental result, which describes the congruence induced by / ® g when the homomorphisms/and g are onto (in which case/ ® g is also onto). As a first consequence, we prove also that A ® B depends only on the largest normal homomorphic images of A and B; namely, A ® B is naturally isomorphic to N(A) ® N(B). In §4, we prove a very peculiar property of the tensor product of semigroups, namely that it preserves consistent monomorphisms (a semigroup homomorphism is consistent if the complement of its image is an ideal or is empty). The fundamental result of §3 can then be extended to consistent morphisms. In §5 we show that the tensor product of semigroups is cokernel preserving in the following sense. Call a sequence A' ■!> A A-A" çoexact if the congruence induced by /' is the smallest in a class in which the image of/is contained. Then the tensor product by any i-indecomposable semigroup X preserves such coexact sequences where/' is onto (or consistent). We also establish the adjoint associativity. In the right exactness result of §5, the assumption on X cannot be lifted, but we show in §6 that this can be fixed by defining another tensor product for semigroups with zero, which is closely related to the tensor product of semigroups and otherwise keeps most of its properties.Our paper is self-contained except that we do not bother to redefine elementary concepts of semigroup theory, which is better done in [1]. Also, the reader will have

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