An intuitive notion of the complexity of an algebra A (eg a group, ring, semigroup, etc) is its rank, ie the minimal size of a generating set. However, if A is uncountable, then any generating set has size |A|, so rank does not tell us anything. All is not lost though, since many other properties of generation can be formulated to distinguish simpler algebras from more complicated ones. Here are two.
Bergman's Property -- for any generating set U of A there is a natural number n such that any element of A can be written as a product (etc) of at most n elements from U. (In other words, the length function is bounded with respect to any generating set.)
Sierpinski rank -- this is defined to be the least integer n (if it exists) such that any countable subset of A is contained in an n-generated subalgebra.
For example, the symmetric group on any infinite set satisfies Bergman's property, and has Sierpinski rank 2. I'll discuss these concepts, and others, in the context of infinite transformation semigroups and partition monoids.
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After the seminar we will take the speaker to lunch.
See the Algebra Seminar web page for information about other seminars in the series.
James East james.east@sydney.edu.au