# The influence of Imre Simon’s work in the theory of automata, languages and semigroups

- 85 Downloads

Imre Simon

## 1 Introduction

Imre Simon, a Brazilian mathematician and computer scientist, was born in Budapest, Hungary on August 14, 1943. He died in São Paulo, Brazil on August 13, 2009, just a day short of his 66th birthday. More details on his life can be found in the preface to the volume written on the occasion of his 60th birthday (Choffrut and Wakabayashi in Theor Inform Appl 39(1): i–vii, 2005). Among the contributions of this volume, the article *Imre Simon: an exceptional graduate student* by Thérien (Theor Inform Appl 39(1):297–304, 2005) deserves special mention. The purpose of this tribute is to review three major results due to Simon that had a considerable influence on the development of automata and semigroup theory.

## 2 \(\mathcal {J}\)-trivial monoids and their languages

Let *A* be a finite alphabet. A language of \(A^*\) is *piecewise testable* if it is a Boolean combination of languages of the form \(A^*a_1A^* a_2 \cdots A^*a_kA^*\), where \(a_1, \ldots , a_k\) are letters. In 1972, Simon obtained an elegant characterization of these languages. Recall that a monoid is \(\mathcal {J}\)-*trivial* if each of its \(\mathcal {J}\)-classes is a singleton.

### Theorem 1.1

(Simon [47, 48]). A language is piecewise testable if and only if its syntactic monoid is finite and \(\mathcal {J}\)-trivial.

This result inspired a lot of subsequent research and a number of alternative proofs have been proposed [1, 2, 23, 25, 28, 29, 60, 63], not to mention unpublished ones. It was proved seven years after Schützenberger’s characterization of star-free languages in terms of aperiodic monoids [45] and these two results, together with Theorem 2.1 below, inspired Eilenberg’s varieties theorem [16].

Theorem 1.1 is a key step in the study of concatenation hierarchies initiated by Cohen and Brzozowski [11]. A detailed account of the numerous developments on this topic, including connections with automata, languages and finite model theory can be found in the survey [39]. For this reason, I will omit this important topic and rather focus on some consequences in semigroup theory. For each integer \(n > 0\), three monoids, \(\mathcal {C}_n\), \(\mathcal {R}_n\) and \(\mathcal {U}_n\) will serve us as examples of \(\mathcal J\)-trivial monoids.

The monoid \(\mathcal {C}_n\) consists of all order preserving and extensive functions from \(\{1, \ldots ,n \}\) to itself. Recall that a transformation *a* on \(\{1, \ldots ,n \}\) is *order preserving* if \(p \leqslant q\) implies Open image in new window and *extensive* if for all *p*, Open image in new window .

*unitriangular*. For example,

*M*

*divides*a monoid

*N*if

*M*is a quotient of a submonoid of

*N*. It is relatively easy to prove the equivalence of Theorems 1.1 and the following result of Straubing [61], but proving one of them is much more difficult.

### Theorem 1.2

*M*, the following conditions are equivalent:

- (1)
*M*is \(\mathcal {J}\)-trivial, - (2)
there exists an integer \(n > 0\) such that

*M*divides \(\mathcal {C}_n\), - (3)
there exists an integer \(n > 0\) such that

*M*divides \(\mathcal {R}_n\), - (4)
there exists an integer \(n > 0\) such that

*M*divides \({\mathcal {U}}_n\).

Almeida obtained another proof of Theorem 1.1 as a byproduct of his complete description of the free profinite \(\mathcal {J}\)-trivial monoid [1]. He proved in particular that the free profinite \(\mathcal {J}\)-trivial monoid on an *n*-letter alphabet is countable and contains \(2^n\) idempotents.

Another important consequence of Theorem 1.1 relates \(\mathcal {J}\)-trivial monoids to ordered monoids. It was first stated and proved by Straubing and Thérien [63]. An *ordered monoid* is a monoid equipped with a partial order \(\leqslant \) compatible with the product: \(x \leqslant y\) implies \(xz \leqslant yz\) and \(zx \leqslant zy\).

### Theorem 1.3

A finite monoid is \(\mathcal {J}\)-trivial if and only if it is a quotient of a finite ordered monoid satisfying the identity \(1 \leqslant x\).

Again, it is not very difficult to establish the equivalence of Theorems 1.1 and 1.3. But Straubing and Thérien gave a direct proof of Theorem 1.3 based on the ideal structure of the monoid and a suitable semigroup expansion. Another proof of Theorem 1.3 proposed in [23] makes use of factorisation trees, a notion inspired by Imre Simon’s factorisation forests described in Sect. 3.

Theorem 1.3, far from being an isolated result, is actually the prototype of similar covering theorems involving ordered monoids. Recall that a *block group* is a monoid in which every regular \(\mathcal {R}\)-class and \(\mathcal {L}\)-class contains a unique idempotent. Many equivalent definitions can be found for instance in [38]. The following result was first stated in [42, Corollary 9.7], but is a consequence of a series of some deep results obtained in the eighties [4, 5, 21, 22, 24, 30]. See [38] for a survey and related results.

### Theorem 1.4

Every finite block group monoid is the quotient of a finite ordered monoid satisfying the identity \(1 \leqslant x^\omega \).

This result can be relativised to monoids with commuting idempotents but, even in this case, it is a challenging open problem to find a direct proof of this result using the techniques proposed in [63] or in [23].

Simon’s original proof [48] relied on the combinatorial study of subwords, one of his favourite topics [58]. He also used the related *shuffle operation* to solve with Ésik an open problem in language theory [17]. Subwords were later used to describe languages whose syntactic monoids are *p*-groups [16, p. 238]. Another elegant proof of Theorem 1.1 via combinatorics on words was proposed by Klíma [28] and is probably the shortest proof known to date.

The combinatorial study of subwords was initiated by Chen, Fox and Lyndon [9] in a different context. See the survey chapter [33, Chapter 6] written by Sakarovitch and Simon, and [43] for applications to Lie algebras.

## 3 Locally trivial semigroups and graph congruences

A semigroup *S* is said to be *locally trivial* if, for every idempotent *e* of *S*, \(eSe = e\). Similarly, a semigroup *S* is said to be *locally idempotent and commutative* if, for every idempotent *e* of *S*, the monoid *eSe* is idempotent and commutative, i.e. a semilattice.

Locally trivial semigroups form a variety of finite semigroups, denoted \(\mathbf {LI}\). If \(\mathbf {V}\) is a variety of finite monoids, let \(\mathbf {V}* \mathbf {LI}\) be the variety generated by all semidirect products of monoids in \(\mathbf {V}\) with semigroups in \(\mathbf {LI}\).

A language of \(A^+\) is *locally testable* if it is a Boolean combination of sets of the form \(uA^*\), \(A^*v\) or \(A^*wA^*\) where \(u, v, w\in A^+\). It it relatively easy to prove that a language is locally testable if and only if its syntactic semigroup belongs to the variety \(\mathbf {SL}* \mathbf {LI}\) where \(\mathbf {SL}\) is the variety of finite semilattices. Quoting Eilenberg [16, p. 222], this is not regarded as a satisfactory solution as it is not clear how to decide membership in this variety. A much better answer was given independently by Brzozowski and Simon [8] and by McNaughton [36], who proved the following result:

### Theorem 2.1

A language is locally testable if and only if its syntactic semigroup is locally idempotent and commutative.

*G*, a

*path congruence*is an equivalence relation on the set of paths of

*G*such that:

- (1)
any two equivalent paths are

*coterminal*(i.e. have the same origin and the same end), - (2)
if

*p*and*q*are equivalent paths, and if*r*,*p*and*s*are consecutive paths, then*rps*is equivalent to*rqs*.

### Theorem 2.2

(Simon [47]) Let \(\sim \) be a path congruence such that, for every pair of loops *p*, *q* around the same state, \(p^2 \sim p\) and \(pq \sim qp\). Then two coterminal paths using the same sets of edges are equivalent.

Theorem 2.2 initiated the graph congruence techniques later developed in a series of subsequent papers by Knast [30], Thérien [64], Thérien-Weiss [66], Straubing [62], Jones and Szendrei [26], Jones and Trotter [27], Almeida [3], Pin, Pinguet and Weil [41], Steinberg [59], etc.

Straubing [62] generalized Theorems 2.1 and 2.2 to describe membership in \(\mathbf {V}* \mathbf {LI}\) in terms of graph congruences. The modern formulation in terms of categories is due to Tilson [67] (see also [44, p. 279, Theorem 4.8.9]).

Theorem 2.2 is also one of the key ingredients for the algebraic proof of McNaughton’s theorem on the equivalence between non deterministic Büchi automata and deterministic Muller automata over infinite words [32].

## 4 Factorization forests

Ramsey’s Theorem is frequently used in combinatorics on words to establish the existence of unavoidable regularities in very long words [33, Chapter 4]. Simon’s factorisation forest theorem is a nested variant of this result. It has become over the years an essential tool in the theory of finite semigroups.

A *factorisation forest* is a function *d* that associates to every word *x* of length \(\geqslant 2\) a factorisation \(d(x) = (x_1, \ldots , x_n)\) of *x* with \(n\geqslant 2\) such that \(x_1, \ldots , x_n\) are nonempty words. The integer *n* is the *degree* of the factorisation.

Let *S* be a finite semigroup and let \(\varphi :A^+\rightarrow S\) be a morphism. A *factorisation forest**d* is \(\varphi \)-*Ramseyan* if, for every word *x* of length \(\geqslant 2\), *d*(*x*) is either of degree 2 or there exists an idempotent *e* of *S* such that \(d(x) = (x_1,\ldots , x_n)\) and \(\varphi (x_1) = \varphi (x_2) = \ldots = \varphi (x_n) = e\) for \(1 \leqslant i \leqslant n\).

Simon first stated his factorisation forest theorem in the technical report [50], which was subsequently published in [52, 53]. He later gave a different proof in [55].

### Theorem 3.1

(Simon [50]) Let \(\varphi :A^+\rightarrow S\) be a morphism onto a finite semigroup. Then there exists a factorisation forest of height \(\leqslant 9|S|\) which is \(\varphi \)-Ramseyan.

The bound 9|*S*| was subsequently improved to 3|*S*| in [13] and to the optimal bound \(3|S|-1\) in [31]. Simon was led to his factorisation forest theorem in connection with his research on three topics: Brown’s theorem on locally finite semigroups, the limitedness problem for the tropical semiring [34, 49, 51, 56] and the star-height problem. Despite their importance, I will not develop further these topics, which are already covered in the historical survey [40].

The term *tropical* was given in honour of Imre Simon, who gave credit to Christian Choffrut in [51], but further investigation among the members of the French school suggests that the term was originally coined by Schützenberger. The term has become standard since the success of *tropical geometry*.

Simon’s factorisation forest theorem has been extended by Colcombet to infinite words [13] and to trees [12]. It is also crucial in the characterization of polynomial languages [42]. Bojanczyck [6] wrote a survey of applications of factorisation forests to fast string algorithms and transducers. There are aspects of Simon’s work that remain to be explored. For instance, very recently an unambiguous version of the Factorization Theorem has been announced in [15, 18].

The reader is referred to the recent survey by Colcombet [14] for more information on Simon’s forest factorisation theorem.

## 5 Conclusion

Due to space constraints, several areas studied by Imre Simon had to be omitted, notably nondeterministic complexity of finite automata [54], compression and entropy [20], string matching algorithms [57]. However, I would like to briefly mention Simon’s research on the Burnside problem for semigroups. Simon’s contribution to this question is reported in the survey [37] coauthored with do Lago.

The general problem concerns the free Burnside monoid satisfying \(x^n = x^{n+m}\), but the results of Green and Rees [19] and McLean [35] essentially reduce the problem for \(n = 1\) to the case \(n = 0\), that is, the Burnside problem for groups. Simon addressed in 1970 what he thought to be the easiest remaining case: the free Burnside monoid satisfying \(x^2 = x^3\). Unfortunately for him, it turns out to be the hardest case, which is not yet completely understood. Simon rediscovered the fact that this monoid was infinite, a result previously obtained by Brzozowski et al. [7], but also extended some properties of the free bands to the free Burnside monoids. He never published his results [46], but later passed the problem on to his best student, Alair Pereira do Lago, who obtained the beautiful results reported in the surveys [37] and [40, Section 6].

Imre Simon’s work had a tremendous impact on subsequent research and considerable influence in semigroup theory, automata, formal languages, combinatorics on words and finite model theory. Ten years after his death, his work is still at the heart of current research.

## Notes

### Acknowledgements

I would like to thank László Márki and Norman Reilly for their help during the preparation of this article.

## References

- 1.Almeida, J.: Implicit operations on finite \(\cal{J}\)-trivial semigroups and a conjecture of I. Simon. J. Pure Appl. Algebra
**69**(3), 205–218 (1991)MathSciNetzbMATHGoogle Scholar - 2.Almeida, J.: Finite Semigroups and Universal Algebra. Series in Algebra, vol. 3. World Scientific, Singapore (1994)zbMATHGoogle Scholar
- 3.Almeida, J.: A syntactical proof of locality of DA. Int. J. Algebra Comput.
**6**(2), 165–177 (1996)MathSciNetzbMATHGoogle Scholar - 4.Ash, C.J.: Finite semigroups with commuting idempotents. J. Aust. Math. Soc. Ser. A
**43**(1), 81–90 (1987)MathSciNetzbMATHGoogle Scholar - 5.Ash, C.J.: Inevitable graphs: a proof of the type II conjecture and some related decision procedures. Int. J. Algebra Comput.
**1**(1), 127–146 (1991)MathSciNetzbMATHGoogle Scholar - 6.Bojańczyk, M.A: Factorization forests, in developments in language theory. In: Diekert, V., Nowotka, D. (eds.) Proceedings of 13th International Conference, DLT 2009, Lecture Notes in Computer Science, vol. 5583 , pp. 1–17. Springer, Berlin (2009)Google Scholar
- 7.Brzozowski, J.A., Čulík II, K., Gabrielian, A.: Classification of noncounting events. J. Comput. Syst. Sci.
**5**, 41–53 (1971)MathSciNetzbMATHGoogle Scholar - 8.Brzozowski, J.A., Simon, I.: Characterizations of locally testable events. Discrete Math.
**4**, 243–271 (1973)MathSciNetzbMATHGoogle Scholar - 9.Chen, K.-T., Fox, R .H., Lyndon, R .C.: Free differential calculus. IV. The quotient groups of the lower central series. Ann. Math.
**68**, 81–95 (1958)MathSciNetzbMATHGoogle Scholar - 10.Choffrut, C., Wakabayashi, Y.: Preface [Imre Simon, the tropical computer scientist]. Theor. Inform. Appl.
**39**(1), i–vii (2005)MathSciNetGoogle Scholar - 11.Cohen, R.S., Brzozowski, J.A.: Dot-depth of star-free events. J. Comput. Syst. Sci.
**5**(1), 1–16 (1971)MathSciNetzbMATHGoogle Scholar - 12.Colcombet, T.: A combinatorial theorem for trees. In: Arge, L., Cachin, C., Jurdzinski, T., Tarlecki, A. (eds.) Proceedings of 34th International Conference on Automata, Languages, and Programming (ICALP). Lecture Notes in Computer Science, vol. 4596, pp. 901–912. Springer, New York (2007)Google Scholar
- 13.Colcombet, T.: Factorisation forests for infinite words. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) Fundamentals of Computation Theory, 16th International Symposium: FCT ’07. Lecture Notes in Comput. Sci. vol. 4639, pp. 226–237. Springer, New York (2007)Google Scholar
- 14.Colcombet, T.: The factorisation forest theorem. In: Pin, J.-É. (ed.) Handbook of Automata Theory, vol. I: Theoretical Foundations, European Mathematical Society, Zürich (2019)Google Scholar
- 15.Dave, V., Gastin, P., Krishna, S.N.: Regular transducer expressions for regular transformations. In: Dawar, A., Grädel, E. (eds.) Proceedings of 33rd IEEE Symposium on Logic in Computer Science, pp. 315–324 (2018)Google Scholar
- 16.Eilenberg, S.: Automata, languages, and machines. Vol. B, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976. With two chapters (“Depth decomposition theorem” and “Complexity of semigroups and morphisms”) by Bret Tilson, Pure and Applied Mathematics, Vol. 59, (1976)Google Scholar
- 17.Ésik, Z., Simon, I.: Modeling literal morphisms by shuffle. Semigroup Forum
**56**(2), 225–227 (1998)MathSciNetzbMATHGoogle Scholar - 18.Gastin, P., Krishna, S.N.: Unambiguous Forest Factorization, CoRR abs/1810.07285 (2018)Google Scholar
- 19.Green, J.A., Rees, D.: On semi-groups in which \(x^r=x\). Proc. Camb. Philos. Soc.
**48**, 35–40 (1952)zbMATHGoogle Scholar - 20.Hansel, G., Perrin, D., Simon, I.: Compression and entropy. In: Finkel, A., Jantzen, M. (eds.) STACS 92, Proceedings of 9th Symposium Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 577, pp. 515–528. Springer, Berlin (1992)Google Scholar
- 21.Henckell, K.: Blockgroups \(=\) powergroups: a consequence of Ash’s proof of the Rhodes type \({\rm II}\) conjecture. In: Monash Conference on Semigroup Theory (Melbourne, 1990), pp. 117–134, World Sci. Publ., River Edge (1991)Google Scholar
- 22.Henckell, K., Margolis, S .W., Pin, J.-É., Rhodes, J.: Ash’s type \({\rm II}\) theorem, profinite topology and Mal’cev products. I. Int. J. Algebra Comput.
**1**(4), 411–436 (1991)zbMATHGoogle Scholar - 23.Henckell, K., Pin, J.-É.: Ordered monoids and \(\cal{J}\)-trivial monoids. In: Algorithmic Problems in Groups and Semigroups (Lincoln. NE, 1998), Trends Mathematicla, pp. 121–137. Birkhäuser Boston, Boston (2000)Google Scholar
- 24.Henckell, K., Rhodes, J.: The theorem of Knast, the \(PG=BG\) and type-\({\rm II}\) conjectures. In: Monoids and Semigroups with Applications (Berkeley, CA, 1989), pp. 453–463. World Scientific Publication, River Edge (1991)Google Scholar
- 25.Higgins, P.M.: A proof of Simon’s theorem on piecewise testable languages. Theoret. Comput. Sci.
**178**(1–2), 257–264 (1997)MathSciNetzbMATHGoogle Scholar - 26.Jones, P.R., Szendrei, M.B.: Local varieties of completely regular monoids. J. Algebra
**150**(1), 1–27 (1992)MathSciNetzbMATHGoogle Scholar - 27.Jones, P.R., Trotter, P.G.: Locality of DS and associated varieties. J. Pure Appl. Algebra
**104**(3), 275–301 (1995)MathSciNetzbMATHGoogle Scholar - 28.Klíma, O.: Piecewise testable languages via combinatorics on words. Discrete Math.
**311**(20), 2124–2127 (2011)MathSciNetzbMATHGoogle Scholar - 29.Klíma, O., Polák, L.: Alternative automata characterization of piecewise testable languages. In: Developments in Language Theory. Lecture Notes in Computer Sciecne, vol. 7907,, pp. 289–300. Springer, Heidelberg (2013)Google Scholar
- 30.Knast, R.: Some theorems on graph congruences. RAIRO Inform. Théor.
**17**(4), 331–342 (1983)MathSciNetGoogle Scholar - 31.Kufleitner, M.: The height of factorization forests. In: Ochmanski, E., Tyszkiewicz, J. (eds.) Proceedings of 33rd Symposium, Mathematical Foundations of Computer Science 2008. Lecture Notes in Computer Science, vol. 5162, pp. 443–454. Springer, Berlin (2008)Google Scholar
- 32.Le Saec, B., Pin, J.-É., Weil, P.: Semigroups with idempotent stabilizers and applications to automata theory. Int. J. Algebra Comput.
**1**, 291–314 (1991)MathSciNetzbMATHGoogle Scholar - 33.Lothaire, M.: Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 17. Cambridge University Press, Cambridge (1983)zbMATHGoogle Scholar
- 34.Mandel, A., Simon, I.: On finite semigroups of matrices. Theoret. Comput. Sci.
**5**(2), 101–111, (1977/78)Google Scholar - 35.McLean, D.: Idempotent semigroups. Am. Math. Mon.
**61**, 110–113 (1954)MathSciNetzbMATHGoogle Scholar - 36.McNaughton, R.: Algebraic decision procedures for local testability. Math. Syst. Theory
**8**(1), 60–76 (1974)MathSciNetzbMATHGoogle Scholar - 37.Pereira Do Lago, A., Simon, I.: Free Burnside semigroups. Theor. Inform. Appl.
**35**(6), (2001) 579–595 (2002). A tribute to Aldo de LucaGoogle Scholar - 38.Pin, J.-É.: \(PG=BG\), a success story. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups (York, 1993). NATO Advanced Science Institutes Series C: Mathematical and Physical Science, vol. 466, pp. 33–47. Kluwer, Dordrecht (1995)Google Scholar
- 39.Pin, J.-É.: The dot-depth hierarchy, 45 years later. In: Konstantinidis, S., Moreira, N., Reis, R., Jeffrey, S. (eds.) The Role of Theory in Computer Science–Essays Dedicated to Janusz Brzozowski, pp. 177–202. Word Scientific (2017)Google Scholar
- 40.Pin, J.-É.: Open Problems about Regular Languages, 35 Years Later. In: Konstantinidis, S., Moreira, N., Reis, R., Jeffrey, S. (eds.) The Role of Theory in Computer Science, Essays dedicated to Janusz Brzozowski, pp. 153–176. Word Scientific (2017)Google Scholar
- 41.Pin, J.-É., Pinguet, A., Weil, P.: Ordered categories and ordered semigroups. Commun. Algebra
**30**, 5651–5675 (2002)MathSciNetzbMATHGoogle Scholar - 42.Pin, J.-É., Weil, P.: Polynomial closure and unambiguous product. Theory Comput. Syst.
**30**(4), 383–422 (1997)MathSciNetzbMATHGoogle Scholar - 43.Reutenauer, C.: Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7. The Clarendon Press, Oxford University Press, New York (1993). Oxford Science PublicationsGoogle Scholar
- 44.Rhodes, J., Steinberg, B.: The \(q\)-theory of Finite Semigroups, Springer Monographs in Mathematics. Springer, New York (2009)zbMATHGoogle Scholar
- 45.Schützenberger, M.-P.: On finite monoids having only trivial subgroups. Inf. Control
**8**, 190–194 (1965)MathSciNetzbMATHGoogle Scholar - 46.Simon, I.: Notes on non-counting languages of order 2. ManuscriptGoogle Scholar
- 47.Simon, I.: Hierarchies of Events with Dot-Depth One, PhD thesis, University of Waterloo, Waterloo, Ontario, Canada, (1972)Google Scholar
- 48.Simon, I.: Piecewise testable events. In: Brackage, H. (ed.) Proceedings of 2nd GI Conference. Lecture Notes in Computer Science vol. 33, pp. 214–222. Springer, Berlin (1975)Google Scholar
- 49.Simon, I.: Limited subsets of a free Monoid. In: Proceedings of 19th Annual Symposium on Foundations of Computer Science, Piscataway, NJ, pp. 143–150. IEEE (1978)Google Scholar
- 50.Simon, I.: Factorization Forests of Finite Height, Rap. Tech., Report 87–73, Laboratoire d’Informatique Théorique et Programmation, Paris, (1987)Google Scholar
- 51.Simon, I.: Recognizable sets with multiplicities in the tropical semiring. In: Chytil, M., Janiga, L., Koubek, V. (eds.) Proceedings of 13th Symposium, Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 324, pp. 107–120. Springer (1988)Google Scholar
- 52.Simon, I.: Properties of factorization forests. In: Proceedings of Formal Properties of Finite Automata and Applications, Ramatuelle, France, Berlin. Lecture Notes in Computer Science, vol. 386, pp. 65–72. Springer (1989)Google Scholar
- 53.Simon, I.: Factorization forests of finite height. Theoret. Comput. Sci.
**72**(1), 65–94 (1990)MathSciNetzbMATHGoogle Scholar - 54.Simon, I.: The nondeterministic complexity of a finite automaton. In: Mots, mélanges offerts à, Schützenberger, M.P., Lothaire, M. (eds.) pp. 384–400. Hermès, Paris, Lang. Raison. Calc. (1990)Google Scholar
- 55.Simon, I.: A short proof of the factorization forest theorem. In: Nivat, M., Podelski, A. (eds.), Tree Automata and Languages (Le Touquet, 1990), Amsterdam. Study Computer Science Artificial Intelligence vol. 10, pp. 433–438. North-Holland (1992)Google Scholar
- 56.Simon, I.: On semigroups of matrices over the tropical semiring. RAIRO Inform. Théor. Appl.
**28**(3–4), 277–294 (1994)MathSciNetzbMATHGoogle Scholar - 57.Simon, I.: String matching algorithms and automata. In: Results and Trends in Theoretical Computer Science (Graz, 1994). Lecture Notes in Computer Science, vol. 812, pp. 386–395. Springer, Berlin (1994)Google Scholar
- 58.Simon, I.: Words distinguished by their subwords. In: Proceedings of WORDS’03. TUCS General Publication vol. 27, pp. 6–13. Turku Center Computer Science, Turku (2003)Google Scholar
- 59.Steinberg, B.: A structural approach to the locality of pseudovarieties of the form \(\mathbf{LH}\,\)Ⓜ\(\,\mathbf{V}\). Internat. J. Algebra Comput.
**20**(2), 307–318 (2010)Google Scholar - 60.Stern, J.: Characterizations of some classes of regular events. Theoret. Comput. Sci.
**35**(1), 17–42 (1985)MathSciNetzbMATHGoogle Scholar - 61.Straubing, H.: On finite \({\cal{J}}\)-trivial monoids. Semigroup Forum
**19**(2), 107–110 (1980)MathSciNetzbMATHGoogle Scholar - 62.Straubing, H.: Finite semigroup varieties of the form \({V}\ast {D}\). J. Pure Appl. Algebra
**36**(1), 53–94 (1985)MathSciNetzbMATHGoogle Scholar - 63.Straubing, H., Thérien, D.: Partially ordered finite monoids and a theorem of I. Simon. J. Algebra
**119**(2), 393–399 (1988)MathSciNetzbMATHGoogle Scholar - 64.Thérien, D.: Two-sided wreath product of categories. J. Pure Appl. Algebra
**74**(3), 307–315 (1991)MathSciNetzbMATHGoogle Scholar - 65.Thérien, D.: Imre Simon: an exceptional graduate student. Theor. Inform. Appl.
**39**(1), 297–304 (2005)MathSciNetzbMATHGoogle Scholar - 66.Thérien, D., Weiss, A.: Graph congruences and wreath products. J. Pure Appl. Algebra
**36**(2), 205–215 (1985)MathSciNetzbMATHGoogle Scholar - 67.Tilson, B.: Categories as algebra: an essential ingredient in the theory of monoids. J. Pure Appl. Algebra
**48**(1–2), 83–198 (1987)MathSciNetzbMATHGoogle Scholar