By Assis de Azevedo (auth.), Jorge Almeida, Gabriela Bordalo, Philip Dwinger (eds.)

This quantity includes papers which, for the main half, are in response to talks given at a global convention on Lattices, Semigroups, and common Algebra that was once held in Lisbon, Portugal throughout the week of June 20-24, 1988. The convention was once devoted to the reminiscence of Professor Antonio Almeida Costa, a Portuguese mathematician who drastically contributed to the advance of th algebra in Portugal, at the 10 anniversary of his loss of life. the topics of the convention replicate a few of his examine pursuits and people of his scholars. the aim of the convention was once to collect best specialists in Lattices, Semigroups, and common Algebra and to advertise a dialogue of modern advancements and traits in those components. All 3 fields have grown quickly over the last few a long time with various levels of interplay. Lattice conception and common Algebra have traditionally advanced along with a wide overlap among the teams of researchers within the fields. extra lately, strategies and ideas of those theories were used greatly within the idea of semigroups. Conversely, a few advancements in that zone may perhaps motivate additional advancements in common Algebra. nevertheless, thoughts of semi crew thought have certainly been hired within the examine of semilattices. a number of papers during this quantity tricky on those interactions.

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**Example text**

3) A ny two (binary) idempotent polynomials of A permute. Proof. As remarked earlier, (1) and (2) force A to be polynomially equivalent to a (faithful) module M over a unital ring R. Two polynomials P(x1"'" Xlll) and q(Y1"'" Yn) of an algebra A permute iff p(q(a l l ,··· ,a 1n ),···, q(a Ill1 ,··· ,allln» for all aij E A, i = 1, ... ,m; j = = q(p(a l l ,··· ,a m1 ),··· ,p(a 11,,··· ,allln » 1, ... ,no But every polynomial function of M (hence of A) is of the form for fixed elements ri E R and fixed a E M.

Then R is a skew field, hence M a vector space over R. For a short ad-hoc proof of this fact, let N(a) = {r E R: ra = O} denote the annihilator of a E M and observe that N(a) =1= N(b) implies Ra = Rb for a,b E M\{O} (because Ra, Rb and R(a+b) are simple). From this, it easily follows that N(a) is the annihilator of the whole module M, and by faithfulness, N(a) = 0 for a=l= O. Hence the multiplication map r -+ ra from R to Ra is an isomorphism, and so R is a skew field. 52 Notice that the previous arguments fail if M itself is a cyclic module.

While Abelian algebras are characterized by a minimality condition for the commutator, so-called neutral algebras are characterized by the maximality condition C( a,(3) == al\/3 for all a,/3 E L. 9. The congruence lattice of a neutral algebra is meet-semidistributive. 6. The interplay between universal algebra and geometry has been demonstrated impressively by Wille [21] and Gumm [7], [8]. The latter author used it as an easy approach to commutator theory. The geometrical point of view will also play some role in the subsequent paragraphs, where we shall construct large classes of neutral algebras.