Content↩︎

1. Potential properties of semigroups, embeddability

2. Varieties: structure problems and finiteness conditions

3. Endomorphisms

4. Solvable and unsolvable classes of finite semigroups and groups

5. Power semigroups

6. Inclusive varieties

7. Partial groupoids (pargoids) and related models

8. Short biography of Evgeniy Sergeevich Lyapin

Introduction↩︎

Evgeniy Sergeevich Lyapin (1914-2005) is a world famous scientist and one of the founders of Semigroup Theory - an important branch of modern algebra. He is the author of the first monograph in this field to appear in the world literature. The central theme of E.S. Lyapin’s research was the composition of transformations, one of the fundamental operations in mathematics.

This mathematical object plays a crucial role in both the exact and the human sciences, including linguistics, as well as in information technology. Lyapin’s research problems and investigations continue to attract interest in both classical areas and in new mathematical theories that have emerged in recent decades.

During the 1970s–1990s, he worked successfully in another branch of algebra — the theory of partial algebraic operations. Together with his student, Professor A. E. Evseev, he published the first monograph in the world literature devoted to this subject.

E. S. Lyapin was also an outstanding teacher and the author of influential textbooks and problem books used by several generations of students. He selected problems for his students with great care, from both scientific and methodological perspectives. More than forty of his graduate students defended their dissertations under his supervision and went on to become well-known specialists.

This notebook presents a number of open problems posed by mathematics - graduates of the Lyapin’s School. We dedicate this collection to the 110th anniversary of the founder of this school, as a tribute to his memory.

1 Potential properties of semigroups, embeddability↩︎

E.S. Lyapin introduced the concept of potential properties of semigroups in his book [1].

We say that a subset \(A\subset S\times S\) (for some semigroup \(S\)) is potentially connected by some predicate \(p\left({x,y}\right)\) in the class of semigroups \(K\) if there exists an oversemigroup \(T\in K\) of \(S\) such that \(p({a,b})\) is true for any pair \(({a,b})\in A\). In this case we say that the subset \(A\) is potentially \(p\) connected.

Lyapin considered in his book the well-known Green’s relations \({\mathfrak L}\) and \({\mathfrak R}\) as predicates and found conditions for the potential \({\mathfrak L}\) and \({\mathfrak R}\) connectedness of subsets in the class of all semigroups.

The concept of potential properties easily extends to predicates of any arity, in particular, to the unary predicate of invertibility.

In his work [2] E.S. Lyapin finds necessary and sufficient conditions for the potential invertibility of an arbitrary element of a semigroup. Lyapin’s criterion is algorithmically verifiable for finite semigroups.

In this regard, it is worth mentioning the work of E.G. Shutov [3] (a student of Lyapin), who proved that if any element of a semigroup \(S\) is potentially invertible, then \(S\) is embeddable to a group. This result makes it possible to find a system of new quasi-identities, different from the quasi-identities found by A.I. Malcev [4], and to resolve a number of open questions in this area.

The paper [5] contains a proof of the well-known result of S.I. Kublanovsky (a student of Prof. Lesohin, himself a student of Lyapin) that the problem of embedding a finite semigroup into a finite completely 0-simple one is algorithmically unsolvable. This result implies a negative solution to Lyapin’s problem concerning the possibility of algorithmically verifying potential properties for Green’s relation \({\mathfrak J}\) in finite semigroups.

In 1966, J. Rhodes posed the following problem: Is there an algorithm that, for an arbitrary finite semigroup \(S\) and for arbitrary two elements \(a,b\in S\), decides whether \(S\) can be embedded into a finite semigroup \(T\) in which the equations \(a=xby\) and \(b=x'ay'\) are solvable, i.e. \(a\) and \(b\) divide each other?

Based on the Kublanovsky result cited above, a negative solution to this problem was found by S. Kublanovsky and M. Sapir in [6]. This led to the following problem:

Potential properties of semigroups include embedding problems into various recursive classes of finite semigroups. A homomorphic image of a subsemigroup \(T\) of a semigroup \(S\) is called a divisor of \(S\) (or factor of \(S\)).

2 Varieties: structure problems and finiteness conditions↩︎

A variety is a class of algebraic systems that can be defined by a set (finite or infinite) of identities. Theory of varieties originates from the classical work of G. Birkhoff Birkh?, published in 1935. E.S. Lyapin and his students paid special attention to problems concerning varieties of semigroups, and the results of their pioneering work had a significant influence on subsequent research in this area.

The first examples of non-finitely based semigroup varieties were obtained by A.P. Biryukov (a student of Lyapin) in 1965 (see Biruk1?). In 1970 in his work Biruk2? he completely described the lattice of varieties of idempotent semigroups, which plays a special role in the theory of semigroup varieties. This result was independently obtained by J.A. Gerhard in G? and C.F Fennemore in F?.

In 1969, in the paper lyapin?, E.S. Lyapin discovered a new method for constructing infinite irreducible sets of balanced identities.

The first result giving a complete description of all covers for important varieties of semigroups, together with their basis and the number of covers was obtained by A.Ya.Aizenshtat (a student of Lyapin) in 1972 in Aisenstat?, see also Aisen2?.

The first example of a finite simple semigroup that does not have a finite basis of identities, was constructed by G.I. Mashevitsky (a student of Lyapin) in Ma1?. He also constructed the first example of a finite 0-simple semigroup that does not have an irreducible basis of identities in the class of completely 0-simple semigroups (see Ma2?). It is easy to see that this example does not have an irreducible basis of identities in the class of all semigroups.

Varieties of semigroups that are finitely approximated relative to certain basic predicates were described by S.I. Kublanovsky (student of Prof. M.M. Lesohin, himself a student of Lyapin) in 1980 in Kubl1?. This result provided a new series of non-locally finite varieties, in which the problems associated with the corresponding predicates - such as predicates of divisibility, entry into ideals, entry into maximal subgroup - are algorithmically solvable.

It should be noted that a complete description of varieties consisting of residually finite semigroups, obtained in the cited work, was independently derived in the works of E.A. Golubov and M.V. Sapir Golubov? and R. Mackenzie McKen1?, McKen2?. Recall that an algebraic system \(A\) is said to be residually finite if for any two distinct elements \(a, b\in A\) there exists a finite system \(F\) and a homomorphism \(f: A\to F\) such that \(f(a) \neq f(b)\).

It should also be noted that for many well-known predicates, S.I. Kublanovsky obtained an algorithmically verifiable descriptions in the language of indicator systems (see Kubl2?, Shevrin?).

The first examples of semigroup varieties with the undecidable equality and divisibility problems were obtained by I.L. Melnichuk (a student of Lyapin) in Melnich?.

The next natural problem was posed by E.S. Lyapin in the early 1970s. Its solution may have useful applications, for example, in automata theory.

We say that a variety satisfies a property locally if any finitely generated semigroup in the variety satisfies this property. The following problem naturally arises.

Let \(RS_n\) denote the variety generated by all \(0\)-simple semigroups satisfying the identity \(x^2 = x^{n+2}\). Such varieties and their subvarieties are called Rees-Sushkevich varieties (see Kubl4?).

A word \(u\) is called blocking (respectively, non-blocking or avoidable) if, for every finite alphabet the set of words over this alphabet that do not contain images of \(u\) under substitutions, is finite (respectively, infinite). These concepts play an important role in the theory of semigroup varieties and in theoretical combinatorics (see Bean?, Zimin?).

A variation of Tarski’s problem on the recursiveness of the set of finite semigroups with a finite basis of identities (the finitely based semigroups) is probably the most famous and intriguing open problem in the theory of semigroup varieties. The following problem represents a special case of this question.

In the following series of problems we consider varieties of ordered semigroups, that is, semigroups equipped with a partial order compatible with the semigroup operation. Such varieties are defined by identities together with identical inequalities. Identical inequalities are often called identical relations of precedence. For example, the semigroup of real numbers on the segment \(S = [0,1]\) with multiplication satisfies the identical inequality \(xy \leq x\).

E.S. Lyapin proposed the problem of studying varieties defined by identities and identical inequalities in the late 1970s.

3 Endomorphisms↩︎

An important characteristic of mathematical structures is their endomorphisms, that is transformations that preserve the structure. The set of all endomorphisms forms a semigroup under the operation of composition. Lyapin and his students paid special attention to the study of this semigroup.

In the work lyapin3?, published in 1966, Lyapin obtained an abstract characterization of the semigroup of endomorphisms for a fairly general class of mathematical structures.

In 1969, in DlabNeum?, V. Dlab and B. Neumann constructed arbitrarily large finite semigroups with only eight endomorphisms. Their question of whether a smaller number could be achieved was answered by S. Kublanovsky (MR734514, 1983), who showed that for semigroups containing at least four elements the minimal possible number of endomorphisms is four. Moreover, there exist infinitely many finite semigroups with exactly four endomorphisms.

Let \(K\) be a class of semigroups. For each natural number \(n\), denote by \(f_K(n)\) the minimal number of endomorphisms among all finite semigroups from \(K\) having at least \(n\) elements. The function \(f_K(n)\) is called the endomorphic function of the class \(K\). A class \(K\) is called endomorphic if its endomorphic function is unbounded. In particular, we are interested in endomorphic varieties of semigroups. The following problems arise concerning endomorphic functions.

In 1969, M.M. Lesokhin (a student of E. S. Lyapin) posed the problem of describing commutative semigroups that have no endomorphisms other than power functions, i.e. such endomorphisms that map each element to a fixed power of this element (Sverd?, problem 1.34). Below we call commutative semigroups whose endomorphisms are all power functions Lesokhin semigroups.

The first nontrivial example of a Lesohin semigroup was constructed by B.K Boguta (a student of Prof. Lesokhin) in the paper Boguta?. Lesokhin’s problem in the class of regular semigroups was solved by A.I. Kuptsov (a student of Prof. Lesokhin) in Kupcov?.

It can be verified by direct calculations that a finitely generated Lesokhin semigroup is either finite with a single idempotent or has no idempotents. Finite Lesokhin semigroups and \(2\)-generated Lesokhin semigroups without idempotents were described by A.A. Borisov in Borisov1? and Borisov2?. The next series of problems was set by A.A. Borisov (a student of Lyapin).

A commutative semigroup is called an E-semigroup if it is isomorphic to the semigroup of all its endomorphisms under pointwise multiplication. For Abelian groups, this concept was introduced in DlabNeum?. In Borisov4?, A.A. Borisov described several classes of finitely generated commutative E-semigroups: finite ones; semigroups without idempotents; cancellative semigroups; and Archimedean semigroups.

It is proved in Borisov4? that for a finitely generated commutative semigroup \(S\) the following conditions are equivalent:

1. the semigroup of all endomorphisms of \(S\) under pointwise multiplication is a free commutative semigroup;

2. the semigroup of all endomorphisms of \(S\) under pointwise multiplication is an infinite monogenic semigroup generated by an identical endomorphism.

In other words \(S\) is a Lesokhin semigroup.

Let \(S\) be a semigroup without idempotents. For elements \(a, b\in S\) we write \(a < b\) if \(a = ab = ba\). It is easy to verify that \(<\) is a strict partial order on \(S\). Hence one can speak about comparable and incomparable elements and about chains in \(S\).

The length of a finite chain is the number of its elements, and each element forms a chain of length \(1\). If \(S\) contains a chain of length \(r\) (\(r\geq 1\)) but no chain of length \(r+1\), we say that the chain rank of \(S\) equals \(r\). If \(S\) contains chains of length \(n\) for all natural numbers \(n\), then the chain rank of \(S\) is said to be infinite.

In his book Semigroups, E.S. Lyapin emphasized that establishing connections between the properties of endomorphism semigroups and the properties of the underlying structured sets is one of the most important directions in semigroup theory. This direction may be viewed as a natural generalization of the ideas of Galois theory.

For a mathematical structure \(A\), denote by \(End(A)\) (respectively \(Aut(A)\)) the semigroup (respectively group) of its endomorphisms (automorphisms) under composition. The sequence \[End(A), End(End(A)), \ldots \quad (\text{respectively } Aut(A), Aut(Aut(A)), \ldots)\] is called the tower of endomorphisms (respectively tower of automorphisms) of \(A\).

4 Solvable and unsolvable classes of finite semigroups and groups↩︎

A class of finite semigroups \(\Phi\) is called recursive (or decidable) if there exists an algorithm that determines whether a given finite semigroup belongs to this class.

Let \(\mathbb{H}, \mathbb{S}, \mathbb{P}\) denote the operators of taking homomorphic images, subsystems and finite direct products, respectively. These operators and their combinations are called Birkhoff operators (see Kubl6?). If \(\mathbb{K}\) is a Birkhoff operator, then \(\mathbb{K}(\Phi)=\{\mathbb{K}(T)|T\in\Phi\}\). For example:
\(\mathbb{S}(\Phi)\) is the class of semigroups embeddable into semigroups from \(\Phi\);
\(\mathbb{H}(\Phi)\) is the class of semigroups that are homomorphic images of semigroups from \(\Phi\);
\(\mathbb{H}\mathbb{S}(\Phi)\) is the class of divisors of semigroups from \(\Phi\);
\(\mathbb{H}\mathbb{S}\mathbb{P}(\Phi)\) is the pseudovariety generated by \(\Phi\).

In general, the recursiveness of a class of semigroups or groups \(\Phi\) does not imply the recursiveness of the class \(\mathbb{K}(\Phi)\) (see Kubl6?, Albert?, Hall1?). This leads to the following problem.

There exist recursive classes \(\Phi\) for which \(\mathbb{K}(\Phi)\) is recursive for every Birkhoff operator \(\mathbb{K}\). Such classes are called completely solvable.

From the results of Ash? and Shein? it follows that the class of finite inverse semigroups is completely solvable. Other examples of completely solvable classes are presented in Kubl6?. In this regard, the following problems arise.

The subset \(\{M\in \mathbb{K}(\Phi)|\mathbb{H}\mathbb{S}\mathbb{P}(M)\subseteq\mathbb{K}(\Phi)\}\) of the class \(\mathbb{K}(\Phi)\) is called the kernel of the class \(\mathbb{K}(\Phi)\) and is denoted by \(Cor(\mathbb{K}(\Phi))\).

5 Power semigroups↩︎

The set of nonempty subsets of a semigroup \(S\), that is the power set \(P(S)\), forms a semigroup under the natural multiplication of subsets. This semigroup is called the power semigroup of \(S\) and is denoted by \(P(S)\).

In his monograph “Semigroups”, E.S. Lyapin emphasized the importance of studying power semigroups from various perspectives. In 1966, at the International Congress of Mathematicians in Moscow, B.M. Schein formulated a series of open problems in semigroup theory. In particular, he posed the following question: Does the isomorphism of power semigroups \(P(S_1) \cong P(S_2)\) imply the isomorphism of the original semigroups \(S_1 \cong S_2\). The same problem was independently posed by Japanese mathematician T. Tamura in Tamura1?. It is easy to see that in the class of groups the Tamura-Schein problem has the positive solution. Tamura conjectured that this problem has the positive solution also in the general case. However, a negative solution in the class of all semigroups was obtained by E. Mogilyanskaya (a student of Lyapin) in 1972. Namely, in Mogil? she constructed an infinite family of infinite semigroups of arbitrary cardinality that are pairwise non-isomorphic but have isomorphic power semigroups. This result is mentioned in the monograph by J. Almeida Almeida?, pp. 357–358.

A systematic study of the properties of power semigroups began in 1967 with the paper Tamura2? by T. Tamura and J. Shafer. Among the results in this direction we mention an unexpected theorem of by S.G. Bershadsky, (a student of prof. M. M. Lesohin, himself a student of Lyapin) proved in Bersh?: ​​“Every semigroup can be embedded into a power semigroup over a free group." It is also known that every commutative semigroup can be embedded into the power semigroup over an Abelian group (see Trnkova?), while not every semigroup can be embedded into the power semigroup over a periodic group (see Bersh+Kubl?).

These results lead to the following problems.

6 Inclusive varieties↩︎

An inclusive variety is a class of semigroups that can be defined by universal formulas of the form \(u \;\hat{\in}\; V\), where \(u\in X^+\) is a word and \(V\subseteq X^+\) is a finite or a countably infinite set of words. Such formulas are called identical inclusions. If \(V\) is a finite set, then the inclusive identity \(u \;\hat{\in}\; V\) is a first-order formula, that is an elementary formula. An inclusive variety that cannot be defined by the first-order formulas is called a nonelementary (or non-axiomatizable) inclusive variety.

Universal formulas of the form \(U = V\), where \(U, V\) are some sets of words (finite or infinite), were called collective identities by E.S. Lyapin. He proved that every system of identical inclusions is equivalent to a system of collective identities and, conversely, every system of collective identities is equivalent to a system of identical inclusions.

These notions were introduced by E.S. Lyapin in a series of papers L1?, L3?, L4? in the 1970s and 1980s. It turned out that in the language of identical inclusions allows a much finer classification of various classes of groups and semigroups than the classical language of identities.

Many natural classes of semigroups, for example, nilsemigroups, nilpotent semigroups, periodic Clifford semigroups, finite groups, and cyclic \(p\)-groups form inclusive varieties; moreover they are nonelementary inclusive varieties. S. Bratchikov (a student of Lyapin) proved in B? that there exist continuum many nonelementary inclusive varieties of Abelian groups.

Identical inclusions constitute a special class of disjunctive identities, or infinite universally closed disjunctions. Therefore, they may find applications in the theory of varieties of group representations (PV?) and in logical geometry (Plotkin?).

Within this framework several natural problems arise. Many of them were proposed by G. Mashevitsky (a student of Lyapin).

We call two semigroups inclusively equivalent if they satisfy the same identical inclusions; equivalently if they generate the same inclusive variety.

A semigroup is said to be determined by identical inclusions up to isomorphism if every semigroup satisfying the same identical inclusions is isomorphic to it.

7 Partial groupoids (pargoids) and related models↩︎

In the 1960s–1980s, E.S. Lyapin developed the theory of partial algebraic operations. The main outcome of this work was the monograph ESL1?, written jointly with his student, Prof. A.E. Evseev.

This research was motivated by the fact that many important classes of transformations arising in different branches of mathematics form partial groupoids (called pargoids in Lyapin’s terminology) with respect to the operation of composition. For example, the class of morphisms of a category, the class of projections, and the class of closure operators form pargoids that are not semigroups.

The important notion of locally finite embeddability (that is, embeddability into a finite semigroup or group), which plays a significant role in the theory of quasivarieties and in solving word equality problems (see Evans?), is naturally formulated in the language of partial groupoids. The unsolvability of a number of well-known algorithmic problems for finite groups and semigroups has been proved using partial groupoids (see HKMST?, Kublan1?, Kublan2?).

The non-axiomatizability of certain classes of semigroups (or algebras of other signatures), for example inclusive varieties, can also be established using partial groupoids together with the Łoś–Tarski theorem (see Malcev?, GM2?).

A semigroup \(S\) is called an outer (respectively, internal) semigroup extension of a pargoid \(P\) if \(P\subseteq S\) (respectively, \(P=S\)) and the restriction of the operation of \(S\) to \(P\) coincides with the operation in \(P\).

A directed endomorphism, that is a projection of an ordered set is called a closure operator on this set.

A class of algebraic systems is called a class of models if its signature contains only relations. A partial binary operation can be represented by a ternary relation, which allows pargoids, groups, and semigroups to be viewed as models. This approach provides new classification possibilities, as demonstrated in the papers Gorn1?, Gorn2? by O.M. Gornostaev (a student of Prof. A.E. Evseev, himself a student of Lyapin).

A model \(M\) with a ternary relation is said to be representable in a groupoid \(A\) (in particular, in a semigroup or a group), if there exists an injective homomorphism from \(M\) into the corresponding ternary model of \(A\). This definition can be generalised to universal algebras of arbitrary signature.

If \(C\) is a variety, then the class of models representable in \(C\) is a quasivariety of models, that is, the representability condition can be expressed by quasiidentities (see Gorn2?).

An involution is a unary operation \(x\to x^{-1}\) that satisfies the identity \((x^{-1})^{-1}=x\). Semigroups equipped with an involution are called involuted semigroups. Clifford semigroups and inverse semigroups are examples of involuted semigroups.

Short biography of Evgeniy Sergeevich Lyapin

"Cast your bread upon the waters, for you will find it after many days." - Ecclesiastes 11:1

Evgeniy Sergeevich Lyapin was an outstanding Russian mathematician, one of the pioneers of the algebraic theory of semigroups, and the author of the world’s first monograph on this subject. The following remarks add several details to Lyapin’s portrait that are less widely known.

A major contribution Russia made to world culture in the twentieth century was the creation of unique scientific schools. Lyapin founded one such school in algebra that became internationally recognized. Graduates of this school later trained generations of mathematics teachers at pedagogical universities across many regions of Russia and other countries.

A defining feature of Lyapin’s character was his integrity and principled conduct. Honesty, kindness, and moral steadfastness often require particular courage, and Lyapin never compromised his conscience for the sake of career advancement. He did not join political organizations—neither the Komsomol nor the Communist Party—and refused to sign collective denunciations. In difficult times he consistently defended students and colleagues who faced unjustified persecution (see some examples in Anna Ya. Aızenshtat and Boris M. Schein Evgeniy Sergeyevich Lyapin—In Memoriam, Aequationes Math. 73 (2007) 1–9).

During World War II, Lyapin refused evacuation from besieged Leningrad together with his institute, despite repeated orders. He remained in the city and endured hunger, blockade, and bombardment. During the war he worked at the Leningrad Branch of the Main Geophysical Observatory, where he took part in calculating the load-bearing capacity of the ice road across Lake Ladoga—the so-called “Road of Life,” which saved the lives of tens of thousands of people. His work in Atmospheric Physics was considered so significant that he was awarded the Order of Lenin and granted the academic title of Professor of Atmospheric Physics.

Lyapin’s development as a scientist was inextricably linked to the traditions of the St. Petersburg mathematical school. His supervisor at the university was Professor Vladimir Abramovich Tartakovskiy, himself a student of the renowned mathematician Boris Nikolaevich Delaunay.

Lyapin defended his PhD thesis in group theory, the theory of reversible transformations. Later he turned to semigroup theory, which studies non-reversible transformations. He explained this choice simply: the vast majority of transformations in nature are not reversible.

Today the importance of semigroup theory is widely recognized. Its methods and results are used not only in classical branches of mathematics but also in automata theory, formal language theory, theoretical computer science, particle physics, and in the study of multidimensional diffusion processes in biological cells. The range of applications of semigroup theory continues to expand.

However, seventy years ago semigroup theory in the USSR was often regarded as a useless and impractical field, and serious pressure was exerted on Lyapin to abandon it. The late 1940s were a difficult period for Soviet fundamental science. By 1949 the campaign for the ideological “purity” of science had reached mathematics. At that time the university administration launched attacks on several areas, including topology, semigroup theory, and functional analysis.

Lyapin was the only participant who spoke publicly in defense of semigroup theory, mathematics and science in general, despite warnings from the administration that the decision had already been made and that he should remain silent. This episode, together with many other events from Lyapin’s life, is described in the book Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups by Christopher Hollings. The attack on Lyapin and two of his colleagues at an open meeting of the Academic Council of Leningrad State University is also described in an article by N. M. Khait, The Pride of Our City, History of Petersburg, No. 1 (23), 2005. At that meeting all those accused remained silent except Lyapin. He stated that the success of science requires the advancement of new ideas. Although history proved him right, at that time he was dismissed from Leningrad University for defending his right to work on semigroup theory. He subsequently continued his work at the Herzen State Pedagogical University of Russia (at that time Leningrad State Pedagogical Institute).

Lyapin devoted a large part of his time and talent to teaching and to improving mathematical education. In particular, he initiated important reforms in the mathematical curricula of pedagogical institutes in the Soviet Union. In the 1960s he served as an expert for UNESCO in India, where he prepared extensive recommendations for improving mathematics education.

In Lyapin’s personality the talents of a mathematician, a teacher, a writer, and a philosopher were harmoniously united. He generated many original mathematical ideas and wrote influential textbooks and monographs. At the same time, he authored several works of fiction and reflected deeply on the development of human civilization in his philosophical book Dynamics of Civilizations. In his writings he discussed the meaning of life, the role of the scientist and the responsibilities of the teacher, and the relationship between pure and applied science. These reflections reveal a deeper understanding of the remarkable personality of Evgeniy Sergeevich Lyapin.

References↩︎