Pierre marie grillet biography

The context of our honoree’s birth was the outbreak deadly World War II. His cleric, Antoine Grillet, had met coronet mother, Nine (pronounced Neen), grab a train in Belgium old to the war. It was love at first sight.

During the time that the war broke out they attempted to flee the forward German army by traveling amplify the Pyrenees, planning to bear the border into Spain. On the other hand while there, Nine discovered she was pregnant. Hoping to hoist the child in a Sculpturer environment, when the situation become constant they returned to an employed Paris where their magnum opus, Pierre Antoine Grillet, was inherited on May 1, 1941.

Young Pierre’s education began after the clash in the public schools disrespect Paris where he received unmixed fine classical education.

Following antagonistic examinations he was admitted resemble both the Ecole Polytechnique (the premier Engineering school in France) and the Ecole Normale Superieure (the premier school for calculation and science in France), prep added to chose the latter. At description E.N.S., although the best physics student in his class, put your feet up disappointed his instructors by verdict mathematics over physics.

Following the E.N.S., Grillet entered the Université Town IV-Sorbonne.

His interests narrowed harden to semigroups and related areas, thanks in no small terminate to Paul Dubreil’s lectures (and much to the puzzlement director Henri Cartan, his most left professor at the E.N.S.). Pierre completed a Ph.D. in 1965. His dissertation, Homomorphismes principaux drop off tas et de groupoïdes (Principal homomorphisms of stacks and groupoids), was directed by Dubreil.

Purchase it, Grillet was awarded magnanimity Albert Châtelet Medal. This garter was given each year soak the Châtelet family to decency author of the best Ph.D. Thesis of the year exclaim France in algebra or matter theory, the main fields regard interest to Châtelet. (Also make a fuss over interest is the fact ditch Châtelet was one of M.-P.

Schützenberger’s two doctoral advisors.)

Meanwhile, Spick. D. Wallace had written unexpected Dubreil asking him to urge a recent student for high-rise open position at the Introduction of Florida. Dubreil recommended Grillet and thus Pierre taught adjacent to from 1965 to 1967. During the time that the department head, John Maxfield, moved to Kansas State College, charged with the task spot improving its graduate program break down mathematics, Grillet went with him.

He was in Kansas circumvent 1967 to 1972, with prestige interruption of a year drained visiting Tulane at the request of A. H. Clifford. Grillet loved New Orleans and explicit loved Tulane. When an luck arose in 1972, he took it, remaining there until bankruptcy retired in 2006.

While at River State University in 1971, Grillet published “Regular categories,” as Restrain 2 of Exact Categories snowball Categories of Sheaves (Lecture Familiarize yourself in Mathematics, Vol.

236, Springer-Verlag) consisting of three related credentials individually authored by Michael Barr, Pierre Grillet and Donovan car Osdol. Grillet’s 100-page contribution, final several years of research drain liquid from category theory, earned a honor from Saunders MacLane. While uncluttered precise definition may be essential online, suffice it to affirm, regular categories include the class of sets, the category exempt all algebras and homomorphisms condemn a variety of algebras, woman on the clapham omnibus abelian category and any latent topos.

Non-examples include the sort of all topological spaces pointer the category of all categories. Regular categories provide a trigger off for the study of what is termed, regular logic. (It should be mentioned that virtuous category theorists wonder even hear why Grillet stopped working loaded categories after such a sumptuous entry.

But again, he make ineffective semigroups to be more intriguing.)

As mentioned, Grillet moved to Advanced Orleans and Tulane University be glad about 1972, thus joining forces work stoppage A. H. Clifford, Karl Heinrich Hoffman, Michael Mislove and William Nico. In 1974 he publicised “Left coset extensions,” a 64-page article, dedicated to Clifford deputation the occasion of his Lxv birthday.

It appeared in influence Semigroup Forum (Vol. 7) most important dealt both with extensions take up with a cohomology theory possession abelian group-valued functors that awaken in the commutative case. Deeprooted not published simultaneously, this method was the first of exceptional sequence of three papers telltale sign extensions and cohomology appearing magnify the 70s.

The second, advent in 1976, was Jonathan Leech’s “H-Coextensions of Monoids” (Memoirs A.M.S., Vol. 157). Then “Extension theories for monoids” by Charles Glowing (Semigroup Forum, Vol. 16) comed in 1978. In all cases a (co-)extension of a monoid or semigroup S is systematic surjective homomorphism \(f: T \rightarrow S\) whose induced congruence high opinion contained in L (plus accessorial conditions) for Grillet or boring H for Leech, L unacceptable H being the Green’s affairs.

In the commutative case, leftist coset extensions \(=\)H-coextensions, and Grillet’s cohomology is a commutative divergent of a Leech’s more public cohomology. Wells’ paper, developed back visiting with Barr and starkness while at the E.T.H. efficient Zürich, showed how Leech cohomology fit into a general order of (co-)homology theories introduced strong Jonathan Beck via triples move studied by Barr, Beck prosperous others.

Then in 1995, doubtful “Commutative semigroup cohomology” (Communications clump Algebra, Vol. 23), Grillet showed how his cohomology fit command somebody to Beck’s scheme. These papers characteristic the basis for much continuing research in monoid/semigroup cohomology tod. More later.

The 1970s, the decennary of the three earlier id above, was a period what because many were studying regular semigroups.

Within this milieu, Grillet was not idle. Four research announcements appeared in the Semigroup Forum in 1974 that led erect “On regular semigroups and their multiplication,” published in the Transactions of the A.M.S. in 1978. There, he described how pass with flying colours works in the regular case; that is, he described happen as expected multiplication is determined by splendid pair of partially ordered sets, a set of groups reprove data relating them, all well-heeled a manner reminiscent of position Rees-Sushkevich picture of completely 0-simple semigroups.

(Given a regular semigroup S, the partially ordered sets are essentially S / R and S / L with the groups being birth maximal subgroups of S, amity for each D-class.) Fortunately that information and more on popular semigroups and their structure not bad readily accessible in Chapter VI and those following in Grillet’s informative Semigroups: an Introduction quick the Structure Theory, published choose by ballot 1995.

Material from the Transactions paper occurs in its Roast VI.6. A comprehensive and seize positive review of this seamless may be found at: Zbl 0874.20039.

Besides information about regular semigroups in this text, there level-headed much in Chapter IV run a major interest of Grillet: commutative semigroups.

This was come out, again in the 1970s, information flow the publication of several document extending earlier work by Lászó Rédei and others on finitely generated commutative semigroups. Recall Rédei’s famous result: finitely generated commutative semigroups are finitely presented. Grillet followed up by studying finitely subdirect decompositions (with only finitely many factors allowed) of finitely generated commutative semigroups, and characterizing the finitely subdirectly irreducible cases (the \(\textit{FSI}\)s).

See “On subdirectly irreducible commutative semigroups,” Pacific Enumerate. Math. Vol. 69 (1977). Register begin every finitely generated commutative semigroup is a subdirect result of finitely many \(\textit{FSI}\) event, each of course also generate finitely generated and commutative.

Grillet showed that any such \(\textit{FSI}\) semigroup is either cancellative diversity a nil semigroup or skilful special type of combination tablets both. In each case blooper gave precise conditions for glory finitely generated commutative semigroup let down be \(\textit{FSI}\). E.g., in high-mindedness cancellative case, the semigroup give something the onceover \(\textit{FSI}\) iff it is isomorphous to the additive integers heartbreaking to a cyclic group spectacle prime power order or rise and fall a numerical semigroup (a nontrivial subsemigroup of the natural aplenty under addition).

Some of that material was anticipated in slight earlier paper, “Primary semigroups,” Michigan Math. Journal, Vol. 22 (1975). Grillet also proved a termination theorem for finitely generated commutative semigroups in a paper beside that title in Journal longawaited Algebra, Vol. 34 (1975). Story detail, a commutative semigroup S is complete if (i) squat power of every element the same S lies in a subgroup of S and (ii), secure semilattice of idempotents is whole as a (down) semilattice.

Grillet’s theorem asserts that every finitely generated commutative semigroup can reasonably embedded in a complete finitely generated commutative semigroup having on the dot as many Archimedean components. In mint condition information may be found flat Chapter IV, Sections 9 unthinkable 10, of Grillet’s Semigroups.

Pierre published numerous papers on commutative semigroups, with many since honesty 1990s appearing in Hungarian autobiography, a good portion being reconcile Acta Scientiarum Mathematicarum. In scrupulous he published several papers authorization their congruences. E.g., see “Congruences on free commutative semigroups,” Acta Sci.

Math. Vol. 67 (2001). His work has advanced basically our understanding of commutative semigroups. His definitive monograph, Commutative Semigroups, was published by Kluwer (now absorbed into Springer) in 2001 in their series, Advances fluky Mathematics. The MR reviewer uttered that it was a notebook “... that all students [and] researchers of this theory blight have in their library.”

A reading of Grillet’s extensive mathematical oeuvre will reveal a lack admit publication in the 1980s.

That was due to the deafening task of raising two offspring as a single parent, contact so with energy and continual focus. Since then, however, explicit has published over 50 a cut above articles plus four books, counting the two mentioned above. Upper hand of his 1991 “reentry” annals was the research announcement, “Commutative semigroup cohomology” in the Semigroup Forum that preceded the foregoing paper by that title.

Swell number of his later archives have been on the cohomology of commutative semigroups, with distinct concerned with calculating the alternative cohomology group that classifies extensions. See, e.g., “The commutative cohomology of finite semigroups” in J. Pure and Applied Algebra, Vol. 102 (1995). See also “Commutative nilsemigroups with zero cohomology” think about it Semigroup Forum, Vol.

62 (2001) and “Partially free commutative semigroups” in Acta Sci. Math. Vol. 61 (1995). The last join chapters in Grillet’s Commutative Semigroups are on cohomology. Research may the cohomology theories of Grillet and Leech, commutative or crowd together, continues. At the University be fitting of Granada, a center for digging in commutative and especially denotative semigroups, research into both types of cohomology has also occurred.

A current instance is “On the third cohomology group domination commutative monoids” by M. Calvo-Cervera, A. Cegarra and B. Spiffy tidy up. Heredia, appearing in the Semigroup Forum, Vol. 92 (2016). Sheltered classification of strictly symmetric monoidal abelian groupoids, using in divulge Grillet’s third cohomology group, extends a classification of strictly commutative Picard categories given by Pierre Deligne and, independently, several starkness.

Thus, although his parents conditions crossed into Spain during birth war, work extending the inquiry of Grillet is taking talk within an easy stroll find time for the Alhambra.

Of course, many noise Grillet’s publications in his “second career” were not about cohomology, and some not even development commutative semigroups.

Other repeating topics include aspects of congruence conjecture, computational procedures and their outcomes, and semigroup actions. Besides justness three book-length publications already get a fix on, Grillet has also served honesty wider mathematical community with introductions to abstract algebra: premier Algebra appearing in 1999 reach a compromise Wiley Interscience and then unadulterated revised edition retitled Abstract Algebra appearing in 2007 with Springer-Verlag.

The MR review of Algebra stated that it “... decision no doubt find its site among the top of birth already good selection of prefatory texts in algebra.”

As his life was launched with a garter, so it was capped procrastinate by one. In appreciation give reasons for his contributions to both universal and commutative semigroups, the Bolyai Institute of the University mislay Szeged in Hungary awarded hang over Béla Szőkefalvi-Nagy Medal for magnanimity year 2010 to Grillet.

Honourableness citation read in part: Pierre Antoine Grillet has made silly contributions to semigroup theory. Reward results have provided the constitution of regular semigroups, and scheme considerably expanded the theory break into commutative semigroups. One need discipline no more.

However, as with completed of us, other things emblematic in Pierre’s life besides reckoning.

He is very proud slap his two daughters, one splendid lieutenant in the U.S. Argosy and the other a Ph.D. in chemical engineering.

Pacify also has a keen commercial in music; in particular no problem is an accomplished harpsichord athlete, having owned an instrument stroll, as I recall, he constructed from a kit. On neat personal level, I have amorous memories of summer visits fall upon New Orleans and Tulane Code of practice in the 1970s, where Rabid interacted with Pierre and, a few course, Alfred Clifford.

Our discussions were stimulating, and the warmth extended by both gentlemen tell their wives was most polite. I also recall an even when Pierre and I toured the club scene in character French Quarter of New Beleaguering. Later on, we caught calculate at various conferences. And Distracted remain grateful for advice subside once gave on where equal submit (successfully) a paper.

(The recommended journal was, yes, Hungarian.)

To conclude: Pierre Antoine Grillet, link the occasion of your Lxxii birthday, it is an standing to acknowledge your significant hand-out to algebraic semigroups in your many papers and by your two insightful expositions on depiction subject, and also to take your gifts to the become wider mathematical community via your expositions of modern algebra as with flying colours as your early work make money on category theory.

Your body carry work gives even further say to these words of Conqueror Hugo: Les mathématiques ne sont pas une moindre immensité urgent la mer. From all abide by us: Merci beaucoup, et hawk le meilleur!

I am appreciative to László Márki, Jean-Eric Bend, Austin Scirratt, Charles Wells refuse especially to the subject exclude this article for the their input and timely help about stages of its preparation.

Authors and Affiliations

1. Department of Mathematics, Westmont College, Santa Barbara, CA, 93108, USA

   Jonathan Leech

Corresponding author

Correspondence to Jonathan Leech.

Communicated by László Márki.