In 1957, N. G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ω contains a free subgroup on 2^card(Ω) generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality ≤ card(Ω), the group Sym(Ω) contains a coproduct of 2^card(Ω) copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras V, and formulated as a statement about functors Set → V. From this one easily obtains analogs of the results stated above with "group" and Sym(Ω) replaced by "monoid" and the monoid Self(Ω) of endomaps of Ω, by "associative K-algebra" and the K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis Ω, and by "lattice" and the lattice Equiv(Ω) of equivalence relations on Ω. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(Ω), Self(Ω) and End_K(V) contains a coproduct of 2^card(Ω) copies of itself.

That paper also gave an example of a group of cardinality 2^card(Ω) that was not embeddable in Sym(Ω), and R. McKenzie subsequently established a large class of such examples. Those results are shown here to be instances of a general property of the lattice of solution sets in Sym(Ω) of sets of equations with constants in Sym(Ω). Again, similar results - this time of varying strengths - are obtained for Self(Ω), End_K(V) and Equiv(Ω), and also for the monoid Rel(Ω), of binary relations on Ω.

Many open questions and areas for further investigation are noted.

Envy is a pan-human phenomenon, universally feared, at least subconsciously, as a particularly dangerous emotion, since it implies hostility and aggression capable of destroying individuals and even societies. Especially in Western society, man has rather successfully repressed his true feelings about envy, which he is taught is the most shameful and reprehensible of all emotions. But even while denying it, man in all cultures has found devices, most but not all of which are symbolic, to cope with his fear of the consequences of envy. In this paper I distinguish between envy and jealousy (the terms are badly confused in English), note the objects that most frequently cause envy (food, children, and health), and analyze envy relationships between both conceptual equals and conceptual nonequals (concluding that one does not "envy down"). I then note the ways in which envy is expressed, including the symbolic "compliment," much feared in many societies because it is recognized as an expression of envy. Particularly striking is the fact that in those situations in which the envy of others is feared, culture dictates a strategy of evasive behavior based on a specific sequence of preferred choices. That is, when an individual fears envy he first attempts to conceal his good fortune; when this is not practical, he falls back on denial that there is reason to envy him; when this is not adequate, he symbolically shares, and only as a last resort does he truly share. This sequence is illustrated with a detailed comparative analysis of food-envy behavior, including tipping, which is explained as a symbolic device to buy off the envy of the waiter. Cultural forms used by an individual to cope with the fear that he is suspected of envy are then noted, as are those used by a person who fears to acknowledge his own envy. Finally, institutional devices to reduce envy are discussed briefly. The paper stresses the ways in which cultural forms have been developed to aid man in coping with psychological problems.

Properties of several sorts of lattices of convex subsets of Rn are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of Rn and the lattice of all convex subsets of Rn-1. The lattices of arbitrary, of open bounded, and of compact convex sets in Rn all satisfy the same identities, but the last of these is join-semidistributive, while for n > 1 the first two are not. The lattice of relatively convex subsets of a fixed set S subset of Rn satisfies some, but in general not all of the identities of the lattice of "genuine" convex subsets of Rn.

Let S = Sym(Omega) be the group of all permutations of an infinite set Omega. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, then there exists a positive integer n such that every element of S may be written as a group word of length at most n in the elements of U. Likewise, if U is a generating set for S as a monoid, then there exists a positive integer n such that every element of S may be written as a monoid word of length at most n in the elements of U. Some related questions and recent results are noted, and a brief proof is given of a result of Ore's on commutators, which is used in the proof of the above result.