Idempotent rank in the endomorphism monoid of a non-uniform partition

We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.Comment: 12 pages; 4 figure

The number of nilpotent semigroups of degree 3

A semigroup is \emph{nilpotent} of degree 3 if it has a zero, every product of 3 elements equals the zero, and some product of 2 elements is non-zero. It is part of the folklore of semigroup theory that almost all finite semigroups are nilpotent of degree 3. We give formulae for the number of nilpotent semigroups of degree 3 with $n\in\N$ elements up to equality, isomorphism, and isomorphism or anti-isomorphism. Likewise, we give formulae for the number of nilpotent commutative semigroups with $n$ elements up to equality and up to isomorphism

FORECAST OF INCOME AND WEALTH FOR THE FARM SECTOR, HOUSEHOLDS, AND THE FARMS THEY OPERATE

Consumer/Household Economics,

FARM INCOME AND THE AGRICULTURAL SECTOR'S CONTRIBUTION TO NATIONAL ECONOMIC OUTPUT

Agribusiness,

MICRO DYNAMICS OF INCOME, DEBT MANAGEMENT AND FARM FINANCIAL PERFORMANCE

Agricultural Finance,

The1997 Scottish referendum : an analysis of the Scottish referendum

Referendums are rare events in the United Kingdom. Only one UK-wide referendum has been held - on membership of the European Community (as it then was) in 1975 - and before 1997 there had been only three other significant referendums: in 1973, in Northern Ireland, on the constitutional position of the province, and in 1979, in Scotland and Wales, on proposals for devolution. Thus the scarcity of cases available for study in itself makes the 1997 referendum on a Scottish parliament worthy of close attention. In addition, however, the fact that Scottish voters were asked to vote on, not one, but two questions - whether or not they were in favour of a Scottish parliament and whether such a parliament should have tax-varying powers - made the Scottish referendum unique among the (admittedly few) referendums that have been held in Britain