Construction of lattice orders on the semigroup ring of a positive rooted monoid

Lattice orders on the semigroup ring of a positive rooted monoid are constructed, and it is shown how to make the monoid ring into a lattice-ordered ring with squares positive in various ways. It is proved that under certain conditions these are all of the lattice orders that make the monoid ring into a lattice-ordered ring. In particular, all of the partial orders on the polynomial ring A[x] in one positive variable are determined for which the ring is not totally ordered but is a lattice-ordered ring with the property that the square of every element is positive. In the last section some basic properties of d-elements are considered, and they are used to characterize lattice-ordered division rings that are quadratic extensions of totally ordered division rings.     

The lattice of semigroup varieties

Semigroups are considered here in terms of the identities which they may satisfy with the emphasis on the identities and on classes of semigroups defined by identities, rather than on the semigroups themselves. These classes, called varieties, form a lattice under inclusion and it is illuminating to interpret properties of semigroup identities and varieties in terms of this lattice, its cardinality, its atoms, its infinite chains, and so on. This paper is an expansion of addresses to the Southeastern Section of the Mathematical Association of America at East Carolina University, March 1968 and to the Symposium on Semigroups and Rings at the University of Puerto Rico, Mayaguez, March 1970. The preparation of the paper was supported in part by NSF Grants GP 6597 and GP 20638.     

The semigroup generated by certain operators on the congruence lattice of a clifford semigroup

Communicated by N. R. Reilly     

The archimedean graph of a positive function on a semigroup

Necessary and sufficient conditions are obtained for the archimedean graph of a positive function on an S-indecomposable semigroup to have no irreducible polygons. Miscellaneous facts about positive functions are obtained. Illustrative examples of positive functions on the free content are given.Some of the results here were announced in Abstract 711-20-44 of Notices AMS. 21 (1974), A-93.     

The lattice of regular subsemigroups of a regular semigroup

It is shown that all regular subsemigroups of an arbitrary regular semigroup form a lattice. The properties of this lattice are investigated.     

The infinitesimal cone of a totally positive semigroup

Given a complex reductive linear algebraic group split over with a fixed pinning, it is shown that all elements of the Lie algebra infinitesimal to the totally positive subsemigroup of lie in the totally positive cone .

     

The lattice of completely simple subsemigroups of a completely simple semigroup

In this paper, we present necessary and sufficient conditions for the lattice of completely simple subsemigroups of a completely simple semigroup to be 0-modular or 0-semidistributive or join semidistributive.     

The Rees algebra of a positive normal affine semigroup ring

Let R be a positive normal affine semigroup ring of dimension d and let be the maximal homogeneous ideal of R. We show that the integral closure of is equal to for all n ∈ℕ with n ≥ d − 2. From this we derive that the Rees algebra R[t] is normal in case that d ≤ 3. If emb dim(R) = d + 1, we can give a necessary and sufficient condition for R[t] to be normal.     

纯正半群上的强同余及其格

给出了纯正半群S的强同余格上同余T的一些判别性质,证明了S上所有基础强同余所构成的集合FCP（S）是CP（S）的完备子格,最后讨论了由纯正半群的正规子半群决定的交完备子格的结构及由“求核”运算确定的（交完备格）同余K的若干性质,还顺带讨论了群同余格.     

On a sublattice of the lattice of congruences on a θ-regular semigroup

Let S be a regular semigroup. The lattice of all idempotent-separating congruences on S and the lattice of all group congruences on S are both modular sublattices of the full lattice of congruences on S. It is evident that the set theoretical union of these two sublattices, (S), is also a sublattice of the full lattice of congruences on S. It is natural to ask: Under what conditions is the sublattice (S) modular? In this paper we obtain a necessary and sufficient condition for the sublattice (S) to be modular when S is what we call a θ-regular semigroup. Bisimple ω-semigroups and simple regular ω-semigroups are θ-regular semigroups and so this paper extends the work of Munn [5] and Baird [1].