Algebra Univers. 76 (2016) 1–30 DOI 10.1007/s00012-016-0380-5 Published online March 15, 2016 © Springer International Publishing 2016 Algebra Universalis
Special elements of the lattice of epigroup varieties
Vyacheslav Yu. Shaprynskiˇı, Dmitry V. Skokov, and Boris M. Vernikov
Abstract. We study special elements of three types (namely, neutral, modular and upper-modular elements) in the lattice of all epigroup varieties. Neutral elements are completely determined (it turns out that only four varieties have this property). We find a strong necessary condition for modular elements that completely reduces the problem of description of corresponding varieties to nilvarieties satisfying identities of some special type. Modular elements are completely classified within the class of commutative varieties, while upper-modular elements are completely determined within the wider class of strongly permutative varieties.
1. Introduction
1.1. Semigroup pre-history. In this article, we study the lattice of all epi- group varieties. Our considerations are motivated by some earlier investiga- tions of the lattice of semigroup varieties and closely related to these investi- gations. We start with a brief explanation of the ‘semigroup pre-history’ of the present work. One of the main branches of the theory of semigroup varieties is an ex- amination of lattices of semigroup varieties (see the survey [16]). If is a V variety, then L( ) stands for the subvariety lattice of under the natural V V order (the class-theoretical inclusion). The lattice operations in L( ) are the V (class-theoretical) intersection denoted by and the join , i.e., the X ∧Y X ∨Y least subvariety of containing both and . V X Y There are a number of articles devoted to examination of identities (first of all, the distributive, modular or Arguesian laws) and some related restric- tions (such as semimodularity or semidistributivity) in lattices of semigroup varieties, and many important results are obtained there. In particular, the semigroup varieties with modular, Arguesian or semimodular subvariety lat- tices were completely classified, and deep results concerning the semigroup varieties with distributive subvariety lattices were obtained. An overview of all these results may be found in [16, Section 11].
Presented by M. Jackson. Received August 2, 2014; accepted in final form February 19, 2015. 2010 Mathematics Subject Classification: Primary: 20M07; Secondary: 08B15. Key words and phrases: epigroup, variety of epigroups, lattice, neutral element, modular element, upper-modular element. The work is supported by Russian Foundation for Basic Research (grant 14-01-00524) and by the Ministry of Education and Science of the Russian Federation (project 2248). 2 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers.
The results mentioned above specify, so to say, ‘globally’ modular or dis- tributive parts of the lattice of semigroup varieties. The following natural step is to examine varieties that guarantee modularity or distributivity, so to say, in their ‘neighborhood’. Saying this, we have in mind special elements in the lattice of semigroup varieties. There are many types of special elements that are considered in lattice theory. Recall some of them. An element x of a lattice L; , is called ∨ ∧ neutral if, for all y, z L, the sublattice of L generated by x, y and z is • ∈ distributive; modular if (x y) z =(x z) y for all y, z L with y z; • ∨ ∧ ∧ ∨ ∈ ≤ upper-modular if (z y) x =(z x) y for all y, z L with y x. • ∨ ∧ ∧ ∨ ∈ ≤ Lower-modular elements are defined dually to upper-modular ones. Note that special elements play an essential role in abstract lattice theory. For instance, if an element x of a lattice L is neutral, then L can be decomposed into subdirect product of its intervals (x]= y L y x and [x)= y L x y (see [2, { ∈ | ≤ } { ∈ | ≤ } Theorem 254 on p. 226]). Thus, the knowledge of which elements of a lattice are neutral gives important information on the structure of the lattice as a whole. All these types of elements as well as some other types of elements of the lattice SEM of all semigroup varieties have been intensively and successfully studied. Briefly, a semigroup variety that is a neutral element of the lattice SEM is called a neutral in SEM variety. An analogous convention is applied to all other types of special elements. Results about special elements in SEM are overviewed in the recent survey [24]. In particular, neutral in SEM varieties were completely determined in [31]; • strong necessary conditions for modular in SEM varieties were discovered • in [6] and [19] (and reproved in a simpler way in [11]); commutative modular in SEM varieties are completely determined in [19]; • commutative upper-modular in SEM varieties were completely classified • in [22]; it is noted in [21] that this result may be extended to strongly permutative varieties without any change (for a definition of strongly per- mutative varieties, see in Subsection 1.2).
1.2. Epigroups. Considerable attention in semigroup theory is devoted to semigroups equipped with an additional unary operation. Such algebras are said to be unary semigroups. As concrete types of unary semigroups, we mention completely regular semigroups (see [8]), inverse semigroups (see [7]), semigroups with involution, etc. One more natural type of unary semigroups is epigroups. A semigroup S is called an epigroup if for any element x of S, there is a natural n such that xn is a group element (this means that xn lies in some subgroup of S). Extensive information about epigroups may be found in the fundamental work [14] by L. N. Shevrin and the survey [15] by the same author. The class of epigroups 2 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 3
The results mentioned above specify, so to say, ‘globally’ modular or dis- is very wide. In particular, it includes all periodic semigroups (because some tributive parts of the lattice of semigroup varieties. The following natural step power of each element in such a semigroup lies in some finite cyclic subgroup) is to examine varieties that guarantee modularity or distributivity, so to say, and all completely regular semigroups (in which all elements are group ele- in their ‘neighborhood’. Saying this, we have in mind special elements in the ments). The unary operation on an epigroup is defined in the following way. lattice of semigroup varieties. There are many types of special elements that If S is an epigroup and x S, then some power of x lies in a maximal subgroup ∈ are considered in lattice theory. Recall some of them. An element x of a lattice of S. We denote this subgroup by Gx. The unit element of Gx is denoted by L; , is called xω. It is well known (see [14], for instance) that the element xω is well de- ∨ ∧ ω ω ω neutral if, for all y, z L, the sublattice of L generated by x, y and z is fined and xx = x x Gx. We denote the element inverse to xx in Gx by • ∈ ∈ distributive; x. The map x x is the just mentioned unary operation on an epigroup −→ modular if (x y) z =(x z) y for all y, z L with y z; S. The element x is called the pseudoinverse of x. Throughout this paper, • ∨ ∧ ∧ ∨ ∈ ≤ upper-modular if (z y) x =(z x) y for all y, z L with y x. we consider epigroups as algebras with the operations of multiplication and • ∨ ∧ ∧ ∨ ∈ ≤ pseudoinversion. This naturally leads to the concept of varieties of epigroups Lower-modular elements are defined dually to upper-modular ones. Note that as algebras with these two operations. The idea to examine epigroups in the special elements play an essential role in abstract lattice theory. For instance, if framework of the theory of varieties was promoted by L. N. Shevrin in [14] an element x of a lattice L is neutral, then L can be decomposed into subdirect (see also [15]). An overview of the first results obtained here may be found product of its intervals (x]= y L y x and [x)= y L x y (see [2, { ∈ | ≤ } { ∈ | ≤ } in [16, Section 2]. The class of all epigroup varieties forms the lattice under Theorem 254 on p. 226]). Thus, the knowledge of which elements of a lattice the class-theoretical inclusion. As usual, operations in this lattice are defined are neutral gives important information on the structure of the lattice as a as follows: the join of two varieties of epigroups and is the least variety whole. X Y that contains both and , and the meet of two varieties of epigroups is All these types of elements as well as some other types of elements of the X Y merely their class-theoretical intersection. lattice SEM of all semigroup varieties have been intensively and successfully If S is a completely regular semigroup (i.e., a union of groups) and x S, studied. Briefly, a semigroup variety that is a neutral element of the lattice ∈ then x is the element inverse to x in the maximal subgroup containing x. Thus, SEM is called a neutral in SEM variety. An analogous convention is applied the operation of pseudoinversion on a completely regular semigroup coincides to all other types of special elements. Results about special elements in SEM with the unary operation traditionally considered on completely regular semi- are overviewed in the recent survey [24]. In particular, groups. We see that varieties of completely regular semigroups (considered as neutral in SEM varieties were completely determined in [31]; • unary semigroups) are varieties of epigroups in the sense defined above. Fur- strong necessary conditions for modular in SEM varieties were discovered • ther, it is well known and may be easily checked that in every periodic epigroup, in [6] and [19] (and reproved in a simpler way in [11]); the operation of pseudoinversion may be expressed in terms of multiplication commutative modular in SEM varieties are completely determined in [19]; • (see [14], for instance). This means that periodic varieties of epigroups may commutative upper-modular in SEM varieties were completely classified • be identified with periodic varieties of semigroups. in [22]; it is noted in [21] that this result may be extended to strongly It seems to be very natural to examine all restrictions on semigroup varieties permutative varieties without any change (for a definition of strongly per- mentioned in Subsection 1.1 for epigroup varieties. This is realized in [26, 28] mutative varieties, see in Subsection 1.2). for identities and related restrictions to the subvariety lattice. In particular, epigroup varieties with modular, Arguesian or semimodular subvariety lattices 1.2. Epigroups. Considerable attention in semigroup theory is devoted to are completely classified and epigroup analogs of results concerning semigroup semigroups equipped with an additional unary operation. Such algebras are varieties with distributive subvariety lattices are obtained there. In the present said to be unary semigroups. As concrete types of unary semigroups, we article, we start with an examination of special elements in the lattice Epi of mention completely regular semigroups (see [8]), inverse semigroups (see [7]), all epigroup varieties. We consider here neutral, modular and upper-modular semigroups with involution, etc. elements in Epi. For brevity, we call an epigroup variety neutral if it is a One more natural type of unary semigroups is epigroups. A semigroup S is neutral element of the lattice Epi. An analogous convention will be applied called an epigroup if for any element x of S, there is a natural n such that xn for all other types of special elements. Our main results give: is a group element (this means that xn lies in some subgroup of S). Extensive information about epigroups may be found in the fundamental work [14] by a complete description of neutral varieties; • L. N. Shevrin and the survey [15] by the same author. The class of epigroups a strong necessary condition for modular varieties; • 4 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers.
a description of commutative modular varieties; • a classification of strongly permutative upper-modular varieties. • We denote by , and the trivial variety, the variety of all semilat- T SL ZM tices and the variety of all semigroups with zero multiplication, respectively. The first main result of the article is the following theorem. Theorem 1.1. For an epigroup variety , the following are equivalent: V (a) is a neutral element of the lattice Epi; V (b) is simultaneously a modular, lower-modular and upper-modular element V of the lattice Epi; (c) coincides with one of the varieties , , , or . V T SL ZM SL∨ZM In contrast, we note that the lattice of completely regular semigroup va- rieties contains infinitely many neutral elements including all band varieties, the varieties of all groups, all completely simple semigroups, all orthodox semi- groups and some other (this readily follows from [17, Corollary 2.9]). Theorem 1.1 is analogous to the description of neutral elements of the lattice SEM obtained by Volkov [31, Proposition 4.1]. To formulate the second result, we need some definitions. We follow the agreement that an adjective indicating a property shared by all semigroups of a given variety is applied to the variety itself; the expressions like ‘com- pletely regular variety’, ‘periodic variety’, ‘nilvariety’, etc. are understood in this sense. A pair of identities wx = xw = w where the letter x does not occur in the word w is usually written as the symbolic identity w = 0. This notation is justified, because a semigroup with such identities has a zero element and all values of the word w in this semigroup are equal to zero. We will refer to the expression w = 0 as a single identity and call such identities 0-reduced. We call an identity u = v substitutive if u and v are plain semigroup words (i.e., they do not contain the operation of pseudoinversion), these words depend on the same letters, and v may be obtained from u by renaming of letters. The second main result of the article is the following theorem. Theorem 1.2. If an epigroup variety is a modular element of the lattice V Epi, then = where is one of the varieties or and is a V M∨N M T SL N nilvariety given by 0-reduced and substitutive identities only. It is easy to see that this theorem completely reduces the problem of the description of modular varieties to nilvarieties defined by substitutive and 0- reduced identities only (see Corollary 3.2 below). Theorem 1.2 is analogous to [19, Theorem 2.5] (that is a slightly stronger version of [6, Proposition 1.6]). Note that Theorem 2.5 of [19] was proved directly and in a simple way in [11]. The third result of the article is the following theorem. Theorem 1.3. A commutative epigroup variety is a modular element of the V lattice Epi if and only if = where is one of the varieties or V M∨N M T 4 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 5 a description of commutative modular varieties; and is a nilvariety that satisfies the commutative law and the identity • SL N a classification of strongly permutative upper-modular varieties. 2 • x y =0. (1.1) We denote by , and the trivial variety, the variety of all semilat- T SL ZM tices and the variety of all semigroups with zero multiplication, respectively. Theorem 1.3 is analogous to the description of commutative semigroup va- The first main result of the article is the following theorem. rieties that are modular elements of the lattice SEM obtained in [19, Theo- rem 3.1]. Theorem 1.1. For an epigroup variety , the following are equivalent: V Let Σ be an identity system written in the language of one associative (a) is a neutral element of the lattice Epi; binary operation and one unary operation. The class of all epigroups that V (b) is simultaneously a modular, lower-modular and upper-modular element satisfy Σ (where the unary operation is treated as pseudoinversion) is denoted V of the lattice Epi; by KΣ. The class KΣ is not obliged to be a variety, because it may not be (c) coincides with one of the varieties , , , or . closed under taking of (infinite) direct products (see [15, Subsection 2.3], for V T SL ZM SL∨ZM instance). Note that the identity systems Σ with the property that K is a In contrast, we note that the lattice of completely regular semigroup va- Σ variety are completely determined in [4, Proposition 2.15]. If K is a variety, rieties contains infinitely many neutral elements including all band varieties, Σ then we use for this variety the standard notation var Σ. It is evident that the varieties of all groups, all completely simple semigroups, all orthodox semi- if the class K consists of periodic epigroups, then it is a periodic semigroup groups and some other (this readily follows from [17, Corollary 2.9]). Σ variety and therefore, is an epigroup variety. Therefore, the notation var Σ is Theorem 1.1 is analogous to the description of neutral elements of the lattice correct in this case. Put = var xm = xm+1, xy = yx for an arbitrary SEM obtained by Volkov [31, Proposition 4.1]. Cm { } natural number m. In particular, = . It will be convenient for us also To formulate the second result, we need some definitions. We follow the C1 SL to assume that = . Recall that an identity of the form agreement that an adjective indicating a property shared by all semigroups C0 T of a given variety is applied to the variety itself; the expressions like ‘com- x1x2 xn = x1πx2π xnπ ··· ··· pletely regular variety’, ‘periodic variety’, ‘nilvariety’, etc. are understood in where π is a non-trivial permutation on the set 1, 2,...,n is called permuta- { } this sense. A pair of identities wx = xw = w where the letter x does not occur tive; if 1π = 1 and nπ = n, then this identity is said to be strongly permutative. in the word w is usually written as the symbolic identity w = 0. This notation A variety that satisfies a [strongly] permutative identity is also called [strongly] is justified, because a semigroup with such identities has a zero element and permutative. The fourth main result of the article is the following theorem. all values of the word w in this semigroup are equal to zero. We will refer to the expression w = 0 as a single identity and call such identities 0-reduced. We Theorem 1.4. A strongly permutative epigroup variety is an upper-modular V call an identity u = v substitutive if u and v are plain semigroup words (i.e., element of the lattice Epi if and only if one of the following holds: they do not contain the operation of pseudoinversion), these words depend on (i) = where is one of the varieties or , and is a nilva- V M∨N M T SL N the same letters, and v may be obtained from u by renaming of letters. The riety satisfying the commutative law and the identity second main result of the article is the following theorem. x2y = xy2; (1.2) Theorem 1.2. If an epigroup variety is a modular element of the lattice V (ii) = m where is an abelian group variety, 0 m 2 and Epi, then = where is one of the varieties or and is a V G∨C ∨N G ≤ ≤ N V M∨N M T SL N satisfies the commutative law and the identity (1.1). nilvariety given by 0-reduced and substitutive identities only. Note that, according to Theorem 1.2, all modular (in particular, all neutral) It is easy to see that this theorem completely reduces the problem of the varieties are periodic. In contrast with this fact, Theorem 1.4 shows that there description of modular varieties to nilvarieties defined by substitutive and 0- exist non-periodic upper-modular varieties. In particular, the variety of all reduced identities only (see Corollary 3.2 below). abelian groups has this property. However, Theorem 1.4 can be considered Theorem 1.2 is analogous to [19, Theorem 2.5] (that is a slightly stronger as an analogue of the description of commutative semigroup varieties that are version of [6, Proposition 1.6]). Note that Theorem 2.5 of [19] was proved upper-modular elements of the lattice SEM given by [22, Theorem 1.2]. directly and in a simple way in [11]. This article consists of five sections. In Section 2, we collect definitions, The third result of the article is the following theorem. notation and auxiliary results used in what follows. Section 3 is devoted to Theorem 1.3. A commutative epigroup variety is a modular element of the the proof of Theorems 1.1–1.4. In Section 4, we collect several corollaries from V lattice Epi if and only if = where is one of the varieties or our main results. Finally, in Section 5, we formulate several open questions. V M∨N M T 6 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers.
2. Preliminaries
2.1. Some properties of the operation of pseudoinversion. It is well known (see [14, 15], for instance) that if S is an epigroup and x S, then ∈ x x = xω. This allows us to replace expressions of the form u u in epigroup identities by uω, for brevity. The following three lemmas are well known and may be easily checked.
Lemma 2.1. The following identity holds in an epigroup S if and only if S is completely regular: x = x (2.1)
Lemma 2.2. If an epigroup variety satisfies the identity xm = xm+1 for V some natural m then the identities xω = x = x = xm hold in . V Lemma 2.3. The following identity holds in an epigroup S if and only if S is a nil-semigroup: x = 0 (2.2)
2.2. Identities of certain varieties. We denote by F the free unary semi- group over a countably infinite alphabet (with the operations and ). Ele- · ments of F are called words. The symbol stands for the equality relation ≡ on the unary semigroup F . If w F , then we denote by c(w) the set of all ∈ letters occurring in w and by t(w) the last letter of w. A letter is called simple [multiple] in a word w if it occurs in w ones [at least twice]. We call a word a semigroup word if it does not include the operation of pseudoinversion. An identity is called a semigroup identity if both its parts are semigroup words. Put = var xy = x2y, x2y2 = y2x2 and ←− = var xy = xy2,x2y2 = y2x2 . P { } P { } The first two claims of the following lemma are well known and may be easily verified; the third one was proved in [1, Lemma 7].
Lemma 2.4. A non-trivial semigroup identity v = w holds: (i) in the variety if and only if c(v)=c(w); SL (ii) in the variety if and only if c(v)=c(w) and every letter from c(v) is C2 either simple both in v and w or multiple both in v and w; (iii) in the variety if and only if c(v)=c(w) and either the letters t(v) and P t(w) are multiple in v and w, respectively, or t(v) t(w) and the letter ≡ t(v) is simple both in v and w.
If w is a semigroup word, then (w) stands for the length of w; otherwise, we put (w)= . We also need the following three remarks about the identities ∞ of nil-semigroups.
Lemma 2.5. Let be a nilvariety. V 6 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 7
2. Preliminaries (i) If the variety satisfies an identity u = v with c(u) = c(v), then also V V satisfies the identity u =0. 2.1. Some properties of the operation of pseudoinversion. It is well (ii) If the variety satisfies an identity of the form u = vuw where the word V known (see [14, 15], for instance) that if S is an epigroup and x S, then vw is non-empty, then also satisfies the identity u =0. ∈ V x x = xω. This allows us to replace expressions of the form u u in epigroup (iii) If the variety satisfies an identity of the form x x x = v and V 1 2 ··· n identities by uω, for brevity. The following three lemmas are well known and (v) = n, then also satisfies the identity x x x =0. V 1 2 ··· n may be easily checked. Proof. (i) and (ii): These are well known and can be easily verified. Lemma 2.1. The following identity holds in an epigroup S if and only if S (iii): If v is a non-semigroup word, it suffices to refer to Lemma 2.3. Now is completely regular: let v be a semigroup word. If (v)
S. Put S∗ = eS. Then S∗ is an epigroup with unit (...,e,...)e E. It e E ∈ follows from (2.3)∈ that the map ε from S into S given by the rule ε(x)= ∗ (...,ex,...)e E is a semigroup homomorphism. It is well known (see [14, 15], ∈ for instance) that an arbitrary semigroup homomorphism ξ from an epigroup S1 into an epigroup S2 is also an epigroup homomorphism (i.e., ξ(a)=ξ(a) for any a S ). Therefore, ε is an epigroup homomorphism from S into S∗. ∈ 1 One can verify that ε separates elements of ES. Let e, f E, x, y S, and ∈ ∈ ε(ex)=ε(fy). Then e ex = e fy and f ex = f fy. Since e, f E, we have · · · · ∈ ex = efy and fex = fy. (2.4) Therefore,
(2.4) (2.4) (2.3) e E (2.4) ex = efy = efex = feex ∈= fex = fy. We see that ex = fy whenever ε(ex)=ε(fy). This means that ε separates elements of ES. Thus, ε and ρ are homomorphisms from S into S∗ and S/ES, respectively, and the intersection of kernels of these homomorphisms is trivial. Therefore, the epigroup S is decomposable into a subdirect product of the epigroups S∗ and S/ES, whence where = var S∗ is a variety generated by a V ⊆M∨N M monoid and = var(S/ES) is a nilvariety. On the other hand, S∗, S/ES , N ∈V whence . We have proved that = . M∨N ⊆V V M∨N Recall that an epigroup is called combinatorial if all its subgroups are trivial. For any natural n, we denote by Cn the n-element combinatorial cyclic monoid. Since Cn is finite, it may be considered as an epigroup. It is well known and may be easily verified that the variety is generated by the epigroup C . Cm m+1 Lemma 2.7. If an epigroup variety is generated by a commutative epigroup M with unit, then = for some abelian group variety and some m 0. M G∨Cm G ≥ Proof. It is well known that the variety of all abelian groups is the least non- periodic epigroup variety. This variety evidently contains the infinite cyclic group. Further, for each natural m, let Gm denote the cyclic group of order m. It is evident that if is periodic, then the set m N Gm has the M { ∈ | ∈ M} greatest element. We denote by G the infinite cyclic group whenever the variety is non-periodic, and the finite cyclic group of the greatest order among all M cyclic groups in otherwise. In both the cases, G . Further, let c be M ∈M m a generator of the monoid Cm. Put X = m N Cm . If the set X { ∈ | ∈ M} does not have the greatest element, then the semigroup m X Cm is not an ∈ epigroup since, for example, no power of the element (...,cm,...)m X belongs ∈ to a subgroup. Therefore, the set of numbers X contains the greatest number. We denote this number by n. Repeating literally arguments from the proof of Theorem 1 in [5], we have that every epigroup from is a homomorphic M image of some subepigroup of the epigroup G Cn. Therefore, = n 1 × M G∨C − where = var G. G 8 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 9
S. Put S∗ = e E eS. Then S∗ is an epigroup with unit (...,e,...)e E. It It is evident that a strongly permutative variety does not contain the vari- ∈ ∈ follows from (2.3) that the map ε from S into S∗ given by the rule ε(x)= eties , , , and ←− . Besides that, every monoid satisfying a permutative LZ RZ P P (...,ex,...)e E is a semigroup homomorphism. It is well known (see [14, 15], identity is commutative. Thus, we have the following corollary of Proposi- ∈ for instance) that an arbitrary semigroup homomorphism ξ from an epigroup tion 2.6 and Lemma 2.7. S1 into an epigroup S2 is also an epigroup homomorphism (i.e., ξ(a)=ξ(a) Corollary 2.8. If is a strongly permutative epigroup variety, then we have for any a S ). Therefore, ε is an epigroup homomorphism from S into S∗. ∈ 1 V One can verify that ε separates elements of ES. Let e, f E, x, y S, and = m where is an abelian group variety, m 0, and is a ∈ ∈ V G∨C ∨N G ≥ N ε(ex)=ε(fy). Then e ex = e fy and f ex = f fy. Since e, f E, we have nilvariety. · · · · ∈ ex = efy and fex = fy. (2.4) 2.4. A direct decomposition of one varietal lattice. We denote by AG Therefore, the variety of all abelian groups. Put (2.4) (2.4) (2.3) e E (2.4) = var x2y = xyx = yx2 =0 . ex = efy = efex = feex ∈= fex = fy. H { } We see that ex = fy whenever ε(ex)=ε(fy). This means that ε separates The aim of this subsection is to prove the following fact. elements of ES. Proposition 2.9. The lattice L( 2 ) is isomorphic to the direct Thus, ε and ρ are homomorphisms from S into S∗ and S/ES, respectively, AG ∨ C ∨H product of the lattices L( ) and L( 2 ). and the intersection of kernels of these homomorphisms is trivial. Therefore, AG C ∨H the epigroup S is decomposable into a subdirect product of the epigroups S∗ Proof. We need two auxiliary statements. and S/ES, whence where = var S∗ is a variety generated by a V ⊆M∨N M Lemma 2.10. If 2 , then = m where is some monoid and = var(S/ES) is a nilvariety. On the other hand, S∗, S/ES , X ⊆ AG ∨ C ∨H X G∨C ∨N G N ∈V abelian group variety, 0 m 2, and . whence . We have proved that = . ≤ ≤ N ⊆H M∨N ⊆V V M∨N Proof. Being a subvariety of the variety 2 , the variety satisfies the Recall that an epigroup is called combinatorial if all its subgroups are trivial. AG ∨ C ∨H X identity x2y = yx2. It is evident that this identity fails in the varieties and For any natural n, we denote by C the n-element combinatorial cyclic monoid. LZ n . Further, Lemma 2.4(iii) and the dual statement imply that this identity Since C is finite, it may be considered as an epigroup. It is well known and RZ n is false in the varieties and ←− as well. Therefore, none of the four mentioned may be easily verified that the variety is generated by the epigroup C . P P Cm m+1 varieties is contained in . Besides that, the variety (and there- X AG ∨ C2 ∨H Lemma 2.7. If an epigroup variety is generated by a commutative epigroup fore, ) satisfies the identity x2yz = x2zy. Substituting 1 for x, we have that M X with unit, then = for some abelian group variety and some m 0. all monoids in are commutative. Proposition 2.6 and Lemma 2.7 imply now m X M G∨C G ≥ that = for some abelian group variety , some m 0, and some X G∨Cm ∨N G ≥ Proof. It is well known that the variety of all abelian groups is the least non- nilvariety . It is evident that . Lemmas 2.1, 2.2, and 2.3 imply that periodic epigroup variety. This variety evidently contains the infinite cyclic N G ⊆ AG m , m, and satisfy the identities (2.1), x = x , and (2.2), respectively. group. Further, for each natural m, let Gm denote the cyclic group of order m. AG C H 2 2 Therefore, the identity x y = x y holds in the variety 2 . But this It is evident that if is periodic, then the set m N Gm has the AG ∨ C ∨H M { ∈ | ∈ M} identity is false in the variety m with m>2. Hence, m 2. This implies greatest element. We denote by G the infinite cyclic group whenever the variety C 2 ≤ 2 2 that the variety m satisfies the identities x y = yx = x y and is non-periodic, and the finite cyclic group of the greatest order among all AG ∨ C ∨H M xyx = xy x. Since 2 , Lemma 2.3 implies that the variety cyclic groups in otherwise. In both the cases, G . Further, let c be N ⊆X ⊆AG∨C2 ∨H2 M ∈M m satisfies the identities x y = xyx = yx = 0, whence . a generator of the monoid Cm. Put X = m N Cm . If the set X N N ⊆H { ∈ | ∈ M} does not have the greatest element, then the semigroup m X Cm is not an Note that if is an arbitrary epigroup variety, then the class of all epigroups ∈ X epigroup since, for example, no power of the element (...,cm,...)m X belongs in satisfying the identity (2.2) is the greatest nilsubvariety of . We denote ∈ X X to a subgroup. Therefore, the set of numbers X contains the greatest number. this subvariety by Nil( ). X We denote this number by n. Repeating literally arguments from the proof Lemma 2.11. Let be an abelian group variety, 0 m 2, and , of Theorem 1 in [5], we have that every epigroup from is a homomorphic G ≤ ≤ N1 N2 M nilvarieties with Nil( m) i for i =1, 2. If 1 = 2, then image of some subepigroup of the epigroup G Cn. Therefore, = n 1 C ⊆N ⊆H N N × M G∨C − where = var G. = . (2.5) G G∨Cm ∨N1 G∨Cm ∨N2 10 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers.
Proof. We say that varieties and differ by an identity u = v if this X1 X2 identity holds in one of the varieties or but fails in another one. Since X1 X2 = , the varieties and differ by some identity. We may assume N1 N2 N1 N2 without loss of generality that this identity holds in but fails in . First, N1 N2 suppose that the identity we mention is a 0-reduced identity u = 0. Since , this identity fails in . In view of Lemma 2.3 and the definition of the N2 ⊆H H variety , there are only two possibilities for the word u: either u x x x H ≡ 1 2 ··· n for some natural n or u x2. Suppose that u x x x for some n. Then ≡ ≡ 1 2 ··· n the identity x x x = 0 holds in but fails in . If m = 2, then the 1 2 ··· n N1 N2 variety contains the variety Nil( )=varx2 =0, xy = yx . The latter N1 C2 { } variety does not satisfy the identity x x x = 0. Therefore, m 1. Then 1 2 ··· n ≤ the variety is completely regular. Lemmas 2.1 and 2.3 imply now G∨Cm that the variety satisfies the identity x x x = x x x . G∨Cm ∨N1 1 2 ··· n 1 2 ··· n But Lemma 2.3 implies that this identity is false in and therefore, in N2 . Thus, (2.5) holds. Now let u x2. Then Lemmas 2.1, 2.2, G∨Cm ∨N2 ≡ and 2.3 imply that the identity x2 = x 2 holds in the variety but G∨Cm ∨N1 is false in the variety . We see again that the inequality (2.5) G∨Cm ∨N2 holds. It remains to consider the case when and differ by some non-0-reduced N1 N2 identity u = v. Suppose that c(u) = c(v). Lemma 2.5(i) then implies that the variety satisfies both the identities u = 0 and v = 0. Then the variety N1 N2 does not satisfy at least one of them, because and do not differ with N1 N2 u = v otherwise. We see that and differ by some 0-reduced identity, N1 N2 and we go to the situation considered in the previous paragraph. Now let c(u)=c(v). Suppose that the identity u = 0 holds in . Since , , H N1 N2 ⊆H we have that the identity u = 0 holds in both the varieties and . Then N1 N2 satisfies also the identity v = 0. But v = 0 fails in , because u = v N1 N2 holds in otherwise. We see that the varieties and differ by some N2 N1 N2 0-reduced identity. This case has already been considered in the previous para- graph. Thus, the identity u = 0 fails in . Analogously, v = 0 fails in . H H A semigroup word w is called linear if each letter from c(w) is simple in w. All non-semigroup words as well as all non-linear semigroup words except x2 equal to 0 in . Since the identity u = v is non-trivial and c(u)=c(v), the H words u and v are linear. Using the fact that c(u)=c(v) again, we have that the identity u = v is permutative. Then it is evident that this identity holds in but fails in . We have proved that the inequality (2.5) G∨Cm ∨N1 G∨Cm ∨N2 holds.
Now we start the direct proof of Proposition 2.9. Let . In V ⊆ AG ∨ C2 ∨H view of Lemma 2.10, = for some abelian group variety , some V G∨Cm ∨N G 0 m 2, and some variety with . Put = . We have that ≤ ≤ N N ⊆H U Cm ∨N = where and . It remains to establish that this V G∨U G ⊆ AG U⊆C2 ∨H decomposition of the variety into the join of some subvariety of the variety V and some subvariety of the variety is unique. AG C2 ∨H 10 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 11
Proof. We say that varieties and differ by an identity u = v if this Let = where and . We need to verify that X1 X2 V G ∨U G ⊆ AG U ⊆C2 ∨H identity holds in one of the varieties or but fails in another one. Since = and = . Let u = v be an arbitrary identity satisfied by . The X1 X2 G G U U G = , the varieties and differ by some identity. We may assume variety satisfies the identity x3 = x4. Then the identity u4v3 = u3v4 holds N1 N2 N1 N2 U without loss of generality that this identity holds in but fails in . First, in the variety . Let us cancel this identity on u3 from the left and on N1 N2 G∨U suppose that the identity we mention is a 0-reduced identity u = 0. Since v3 from the right, thus concluding that u = v holds in Gr( ). There- G∨U , this identity fails in . In view of Lemma 2.3 and the definition of the fore, Gr( ) . The opposite inclusion is evident. Thus, Gr( )= . N2 ⊆H H G∨U ⊆G G∨U G variety , there are only two possibilities for the word u: either u x1x2 xn Analogously, Gr( )= . We see that H 2 ≡ ··· G ∨U G for some natural n or u x . Suppose that u x1x2 xn for some n. Then ≡ ≡ ··· = Gr( ) = Gr( ) = Gr( )= . the identity x1x2 xn = 0 holds in 1 but fails in 2. If m = 2, then the G G∨U V G ∨U G ··· N 2 N variety contains the variety Nil( )=varx =0, xy = yx . The latter It remains to check that = . Recall that = m where 0 m 2 N1 C2 { } U U U C ∨N ≤ ≤ variety does not satisfy the identity x x x = 0. Therefore, m 1. Then and , while 2 . It is evident that the variety 2 (and 1 2 ··· n ≤ N ⊆H U ⊆C ∨H C ∨H the variety is completely regular. Lemmas 2.1 and 2.3 imply now therefore, ) is combinatorial. Thus, Lemma 2.10 implies that = k G∨Cm U U C ∨N that the variety satisfies the identity x x x = x x x . for some 0 k 2 and some variety with . G∨Cm ∨N1 1 2 ··· n 1 2 ··· n ≤ ≤ N N ⊆H But Lemma 2.3 implies that this identity is false in and therefore, in Suppose that m = k. Without loss of generality, we may assume that N2 . Thus, (2.5) holds. Now let u x2. Then Lemmas 2.1, 2.2, m
1) deg( ) n; V ≤ 2 2) var x = x1x2 xn+1 =0, xy = yx ; V { ··· } 3) satisfies an identity of the form V x1 xn = x1 xi 1 xi xj xj+1 xn (2.6) ··· ··· − · ··· · ··· for some i and j with 1 i j n. ≤ ≤ ≤ Proposition 2.12 readily implies the following assertion.
Corollary 2.13. If and are arbitrary epigroup varieties, then the follow- X Y ing equality holds: deg( )=min deg( ), deg( ) . X ∧Y X Y The following corollary may be proved quite analogously to Corollary 2.13 of [22] by referring to Proposition 2.12 rather than Proposition 2.11 of [22].
Corollary 2.14. If is an arbitrary epigroup variety and is a nilvariety, V N then the following equality holds: deg( ) = max deg( ), deg( ) . V∨N V N Note that the analog of Corollary 2.14 for arbitrary epigroup varieties is wrong even in the periodic case. For instance, it is easy to deduce from Lemma 2.4(iii), the dual fact, and Proposition 2.12 that deg( ) = deg ←− =2 P P but deg ←− = 3. P∨P Proposition 2.12 and Lemma 2.1 easily imply the following fact. Corollary 2.15. If is an arbitrary epigroup variety and is a completely V X regular variety, then deg( ) = deg( ). V∨X V 2.6. Some properties of special elements of lattices. An element of a lattice L is called distributive if x (y z)=(x y) (x z) for all y, z L. ∨ ∧ ∨ ∧ ∨ ∈ Codistributive elements are defined dually. The following claim is well known. It may be obtained by a combination of [3, Lemma 1 in Section 1 of Chapter II], the dual statement and [2, Theorem 255 on p. 228].
Lemma 2.16. An element of a lattice L is neutral in L if and only if it is distributive, codistributive, and modular in L.
The following assertion is a very special case of [10, Corollary 2.1].
Lemma 2.17. Let L be a lattice with 0 and a an atom which is a neutral element of the lattice L. Then x L is an [upper-, lower-]modular element in ∈ L if and only if x a has the same property. ∨ 2.7. and are atoms. It is evident that the atoms of the lattice Epi SL ZM coincide with the atoms of the lattice SEM. The list of the atoms of the latter lattice is generally known (see [16], for instance). In particular, the following is valid.
Lemma 2.18. The varieties and are the atoms of the lattice Epi. SL ZM 12 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 13
1) deg( ) n; 3. Main results V ≤ 2 2) var x = x1x2 xn+1 =0, xy = yx ; V { ··· } 3) satisfies an identity of the form This section contains the proofs of all four theorems. It is divided into five V subsections. We start with a verification of a special case of Theorem 1.1, x1 xn = x1 xi 1 xi xj xj+1 xn (2.6) ··· ··· − · ··· · ··· namely the claim that the variety is neutral (Subsection 3.1). This fact is SL for some i and j with 1 i j n. used in the proofs of Theorems 1.1, 1.3, and 1.4. After that, we prove The- ≤ ≤ ≤ orem 1.2 (Subsection 3.2). We start with this theorem, because we need it Proposition 2.12 readily implies the following assertion. to verify Theorem 1.1. Since we have started to examine modular varieties, Corollary 2.13. If and are arbitrary epigroup varieties, then the follow- we consider them also in the following subsection. Namely, in Subsection 3.3, X Y ing equality holds: deg( )=min deg( ), deg( ) . we verify Theorem 1.3. After that, we prove Theorems 1.1 and 1.4 in Subsec- X ∧Y X Y tions 3.4 and 3.5, respectively. The following corollary may be proved quite analogously to Corollary 2.13 of [22] by referring to Proposition 2.12 rather than Proposition 2.11 of [22]. 3.1. The variety is neutral. Here we prove the following fact. SL Corollary 2.14. If is an arbitrary epigroup variety and is a nilvariety, V N Proposition 3.1. The variety is a neutral element of the lattice Epi. then the following equality holds: deg( ) = max deg( ), deg( ) . SL V∨N V N Proof. In view of Lemma 2.16, it suffices to verify that the variety is SL Note that the analog of Corollary 2.14 for arbitrary epigroup varieties is distributive, codistributive, and modular. Let and be arbitrary epigroup X Y wrong even in the periodic case. For instance, it is easy to deduce from varieties. Lemma 2.4(iii), the dual fact, and Proposition 2.12 that deg( ) = deg ←− =2 P P Distributivity. We need to verify the inclusion but deg ←− = 3. P∨P ( ) ( ) ( ), Proposition 2.12 and Lemma 2.1 easily imply the following fact. SL∨X ∧ SL∨Y ⊆ SL ∨ X ∧Y because the opposite inclusion is evident. Suppose that an identity u = v Corollary 2.15. If is an arbitrary epigroup variety and is a completely holds in ( ). In particular, it holds in , whence c(u)=c(v) by V X SL ∨ X ∧Y SL regular variety, then deg( ) = deg( ). Lemma 2.4(i). Let u w ,w ,...,w v be a deduction of this identity from V∨X V ≡ 0 1 n ≡ the identities of the varieties and . Further considerations are given by X Y 2.6. Some properties of special elements of lattices. An element of a induction on n. lattice L is called distributive if x (y z)=(x y) (x z) for all y, z L. ∨ ∧ ∨ ∧ ∨ ∈ Induction base. If n = 1, then the identity u = v holds in one of the Codistributive elements are defined dually. The following claim is well known. varieties or . Whence, it holds in one of the varieties or , X Y SL∨X SL∨Y It may be obtained by a combination of [3, Lemma 1 in Section 1 of Chapter II], and therefore ( ) ( ) satisfies u = v. the dual statement and [2, Theorem 255 on p. 228]. SL∨X ∧ SL∨Y Induction step. Now let n>1. Consider the words w1 ,...,wn 1 obtained − from the words w1,...,wn 1, respectively, by equating all the letters that do Lemma 2.16. An element of a lattice L is neutral in L if and only if it is − distributive, codistributive, and modular in L. not not occur in u to some letter in c(u). Clearly, the sequence of words u, w1 ,...,wn 1,v is also a deduction of the identity u = v from the identities − The following assertion is a very special case of [10, Corollary 2.1]. of the varieties and . Thus, we may assume that c(w1),...,c(wn 1) X Y − ⊆ c(u)=c(v). If c(w )=c(w )= = c(w ), then the sequence w ,w ,...,w Lemma 2.17. Let L be a lattice with 0 and a an atom which is a neutral 0 1 ··· n 0 1 n is a deduction of the identity u = v from the identities of the varieties element of the lattice L. Then x L is an [upper-, lower-]modular element in SL∨X ∈ and , and we are done. Suppose now that c(w ) = c(w ) for some L if and only if x a has the same property. SL∨Y k k+1 ∨ 0 k n 1. Let i be the least index with c(w ) = c(w ) and j be the ≤ ≤ − i i+1 greatest index with c(wj) = c(wj 1). Suppose that i>0. Then c(wi)= 2.7. and are atoms. It is evident that the atoms of the lattice Epi − SL ZM c(u)=c(v) and consequences of words w ,w ,...,w and w ,w ,...,w are coincide with the atoms of the lattice SEM. The list of the atoms of the latter 0 1 i i i+1 n deductions of the identities u = w and w = v, respectively, from the identities lattice is generally known (see [16], for instance). In particular, the following i i of the varieties and . Lemma 2.4(i) implies now that the identities u = w is valid. X Y i and w = v hold in the variety ( ). By induction assumption, i SL ∨ X ∧Y Lemma 2.18. The varieties and are the atoms of the lattice Epi. these identities hold also in the variety ( ) ( ). Whence, the SL ZM SL∨X ∧ SL∨Y 14 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers. last variety satisfies the identity u = v too. The case when j ω ω ω ω ω ω ω u, uu ,w1u ,...,wn 1u , vu , vw1 ,...,vwn 1, vv ,v − − is a deduction of the identity u = v from the identities of the varieties SL∨X and . Hence, this identity holds in ( ) ( ). SL∨Y SL∨X ∧ SL∨Y Codistributivity. In view of Lemma 2.18, if is an arbitrary epigroup W variety, then either or = . We need to verify that W ⊇ SL W ∧ SL T ( )=( ) ( ). SL ∧ X ∨Y SL∧X ∨ SL∧Y Clearly, each part of this equality coincides with whenever at least one SL of the varieties or contains . It remains to verify that if X Y SL X SL and , then . This claim immediately follows from the Y SL X ∨Y SL fact that there is a non-trivial identity u = v such that an epigroup variety does not contain the variety if and only if satisfies the identity W SL W u = v (in particular, the identity (xωyωxω)ω = xω has such property, see [15, Corollary 3.2], for instance). Modularity. Let . We need to verify that X ⊆Y ( ) ( ) , SL∨X ∧Y ⊆ SL∧Y ∨X because the opposite inclusion is evident. If , then each part of the in- SL⊆Y clusion evidently coincides with . Now let . Then Lemma 2.18 SL∨X SL Y implies that = , whence ( ) = . Suppose that an iden- SL∧Y T SL∧Y ∨X X tity u = v holds in the variety . It suffices to verify that this identity holds X in ( ) , too. If c(u)=c(v), then u = v holds in by Lemma 2.4(i). SL∨X ∧Y SL Whence it is satisfied in , and we are done. Now let c(u) = c(v). SL∨X Since , Lemma 2.4(i) implies that satisfies an identity s = t with SL Y Y c(s) = c(t). Without loss of generality, we may assume that there is a letter 14 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 15 last variety satisfies the identity u = v too. The case when j 3.4. Neutral varieties. This subsection is devoted to the proof of Theo- rem 1.1. Proof of Theorem 1.1. The implication (a) (b) is evident. ⇒ (b) (c): Since the variety is modular, we may apply Theorem 1.2 and ⇒ V conclude that = where is one of the varieties or and V M∨N M T SL N is a nilvariety. It remains to verify that is one of the varieties or . N T ZM By Lemma 2.18, we have to check that . In other words, we need N ⊆ ZM to verify that is a variety of degree 2. Arguing by contradiction, we N ≤ suppose that is a variety of degree > 2. The variety is lower-modular and N V upper-modular. Corollary 3.2 implies that the variety is lower-modular and N upper-modular, too. A variety is called 0-reduced if it may be given by 0-reduced identities only. One can verify that the variety is 0-reduced. Arguing by contradiction, N we suppose that this is not the case. The ‘only if’ part of the proof of The- orem 3.1 in [31] implies that there is a periodic group variety such that G Nil( ) . Put = Nil( ). Since and the variety is G∨N ⊃N N G∨N N ⊆N N lower-modular, we have =( ) =( ) = = , N G∨N ∧N G∧N ∨N T ∨N N contradicting the claim that . N ⊂N Further, is periodic, whence it may be considered as a semigroup variety. N Therefore, is an upper-modular element of the lattice Per. A semigroup N variety is called proper if it differs from the variety of all semigroups. It follows from [23, Theorem 1] that if a proper semigroup variety of degree > 2 is upper-modular in SEM, then it is commutative. All varieties that appear in the proof of this fact are periodic. So, we have that a variety of degree > 2 is commutative whenever it is an upper-modular element in Per. In particular, the variety is commutative. N Thus, is a 0-reduced and commutative nilvariety. Therefore, . N N ⊆ ZM (c) (a): It is well known that the set of all neutral elements of a lattice L ⇒ forms a sublattice in L (see [2, Theorem 259 on p. 230]). In view of Proposi- tion 3.1 and the equality = , it remains to verify that the variety T SL∧ZM is neutral. By Lemma 2.16, it suffices to check that is distributive, ZM ZM codistributive, and modular. Let and be arbitrary epigroup varieties. X Y Distributivity. We need to verify that ( ) ( ) ( ), ZM ∨ X ∧ ZM ∨ Y ⊆ ZM ∨ X ∧Y because the opposite inclusion is evident. Suppose that an identity u = v holds in ( ). We aim to check that this identity is satisfied by the ZM ∨ X ∧Y variety ( ) ( ). The identity u = v holds in and there ZM∨X ∧ ZM ∨ Y ZM is a deduction of this identity from the identities of the varieties and . X Y 18 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 19 verified by the same arguments as in the proof of the ‘if’ part of Theorem 1 In other words, there are words u ,u ,...,u such that u u, u v, and 0 1 n 0 ≡ n ≡ in [27]. Theorem 1.3 is proved. for each i =0, 1,...,n 1, the identity ui = ui+1 holds either in or in . − X Y Let u0,u1,...,un be the shortest sequence of words with these properties. If 3.4. Neutral varieties. This subsection is devoted to the proof of Theo- all the words u ,u ,...,u are not letters, then u = u = = u holds in 0 1 n 0 1 ··· n rem 1.1. . This means that the sequence of words u ,u ,...,u is a deduction of ZM 0 1 n Proof of Theorem 1.1. The implication (a) (b) is evident. the identity u = v from the identities of the varieties and , ⇒ ZM∨X ZM ∨ Y (b) (c): Since the variety is modular, we may apply Theorem 1.2 and whence u = v holds in ( ) ( ). Now let i be an index such ⇒ V ZM∨X ∧ ZM ∨ Y conclude that = where is one of the varieties or and that ui x for some letter x. Clearly, 0 3.5. Upper-modular varieties. Here we verify Theorem 1.4. To do this, we need several auxiliary statements. Lemma 3.6. If a strongly permutative epigroup variety is an upper-modular V element of the lattice Epi, then is commutative. V Proof. In view of Corollary 2.8, = where is an abelian group V G∨Cm ∨N G variety, m 0, and is a nilvariety. If deg( ) 2, then , and we are ≥ N V ≤ N ⊆ ZM done. Now let deg( ) > 2. By Proposition 2.12, contains the variety = 2 V V X var x = xyz =0, xy = yx . Suppose that is not commutative. Let be a { } V G non-abelian group variety. Since is strongly permutative, every group in is V V abelian. Therefore, the variety ( ) is commutative. Since and G ∧V ∨X X ⊆V the variety is upper-modular, we have that ( ) =( ) . We V G ∧V ∨X G ∨X ∧V see that the variety ( ) is commutative. Hence, there is a deduction G ∨X ∧V of the identity xy = yx from the identities of the varieties and . In G ∨X V particular, there is a word v such that v xy and the identity xy = v holds ≡ either in or in . The claims (i) and (iii) of Lemma 2.5 imply that a G ∨X V variety with the identity xy = v is either commutative or a variety of degree 2. The variety is neither commutative (because is non-abelian) ≤ G ∨X G nor a variety of degree 2 (because deg( ) > 2). Since deg( ) > 2, we have ≤ X V that is commutative. V It is proved in [22, Theorem 1.1] that if is a proper upper-modular in V SEM variety, then first, is periodic (note that this is a very special case V of [12, Theorem 1]), and second, every nilsubvariety of is commutative and V satisfies the identity (1.2). As we have already mentioned in Subsection 1.2, the epigroup analog of the first claim is not true. Our next step is the following partial epigroup analog of the second claim. Proposition 3.7. If a strongly permutative epigroup variety is an upper- V modular element of the lattice Epi, then every nil-semigroup in satisfies the V identity (1.2). Proof. According to Lemma 3.6, the variety is commutative. If all nil-semi- V groups in are singletons, then the desirable conclusion is evident. Suppose V now that contains a non-singleton nil-semigroup N. Let be the vari- V N ety generated by N. Clearly, this variety is commutative. It is evident that . We need to verify that satisfies the identity (1.2). Put ZM⊆N N = var x2y = xy2, xy = yx, x2yz =0 I { } and = . It is clear that , whence . N N ∧I ZM⊆I ZM⊆N A semigroup analog of the proposition that we are verifying here is proved in [22] (see the last paragraph of Section 3 in that article). The arguments used there are based on the fact that there is a variety such that the following X two claims are valid: (i) ( ) ; X ∨N ∧V ⊆I 20 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 21 3.5. Upper-modular varieties. Here we verify Theorem 1.4. To do this, (ii) if v x2y, xyx, yx2 and w xy2,yxy,y2x , then the identity v = w ∈{ } ∈{ } we need several auxiliary statements. fails in . X Lemma 3.6. If a strongly permutative epigroup variety is an upper-modular In [22], a certain periodic group variety plays the role of . Here we should V X element of the lattice Epi, then is commutative. take another . Namely, put = where V X X LZM ∨ RZM = var xyz = xy and = var xyz = yz . Proof. In view of Corollary 2.8, = m where is an abelian group LZM { } RZM { } V G∨C ∨N G variety, m 0, and is a nilvariety. If deg( ) 2, then , and we are The variety satisfies the identity xyzxy = xy. Therefore, Lemma 2.4(i) ≥ N V ≤ N ⊆ ZM X done. Now let deg( ) > 2. By Proposition 2.12, contains the variety = implies that . Further, substituting 1 for x and y in the identity 2 V V X SL X var x = xyz =0, xy = yx . Suppose that is not commutative. Let be a xyzxy = xy, we obtain that all groups in are singletons. Hence every { } V G X non-abelian group variety. Since is strongly permutative, every group in is commutative semigroup in is a nil-semigroup. Further, satisfies the iden- V V 2 X X abelian. Therefore, the variety ( ) is commutative. Since and tity xy =(xy) , whence all nil-semigroups in lie in by Lemma 2.5(ii). G ∧V ∨X X ⊆V X ZM the variety is upper-modular, we have that ( ) =( ) . We Since the variety is commutative, . The variety is upper- V G ∧V ∨X G ∨X ∧V X ∧V X ∧ V ⊆ ZM V see that the variety ( ) is commutative. Hence, there is a deduction modular and . Therefore, G ∨X ∧V N ⊆V of the identity xy = yx from the identities of the varieties and . In G ∨X V ( ) =( ) = . particular, there is a word v such that v xy and the identity xy = v holds X ∨N ∧V X ∧V ∨N ⊆ ZM ∨ N N ⊆I ≡ We have proved claim (i). To verify claim (ii), we note that if satisfies a either in or in . The claims (i) and (iii) of Lemma 2.5 imply that a G ∨X V X variety with the identity xy = v is either commutative or a variety of degree semigroup identity v = w, then the words v and w have the same prefix of length 2 and the same suffix of length 2. Clearly, this is not the case whenever 2. The variety is neither commutative (because is non-abelian) ≤ G ∨X G 2 2 2 2 nor a variety of degree 2 (because deg( ) > 2). Since deg( ) > 2, we have v x y, xyx, yx and w xy ,yxy,y x . Now we can complete the proof ≤ X V ∈{ } ∈{ } that is commutative. by using the same arguments as in the last paragraph of [22, Section 3]. V It is proved in [22, Theorem 1.1] that if is a proper upper-modular in The proof of the following statement repeats almost literally the ‘only if’ V SEM variety, then first, is periodic (note that this is a very special case part of the proof of Theorem 2 in [27]. V of [12, Theorem 1]), and second, every nilsubvariety of is commutative and Proposition 3.8. If a nilvariety of epigroups satisfies the identities (1.2) V X satisfies the identity (1.2). As we have already mentioned in Subsection 1.2, and xy = yx, then is an upper-modular element of the lattice Epi. the epigroup analog of the first claim is not true. Our next step is the following X Proof. It is easy to prove (see [27, Lemma 2.7], for instance) that satisfies partial epigroup analog of the second claim. X the identity x2yz = 0. Thus, . Put X ⊆I Proposition 3.7. If a strongly permutative epigroup variety is an upper- 2 3 2 V U = x ,x ,x y, x1x2 xn n N . modular element of the lattice Epi, then every nil-semigroup in satisfies the { ··· | ∈ } V identity (1.2). It is evident that any subvariety of may be given in only by identities of I I the type u = v or u = 0 where u, v U. Lemma 2.5 implies that if u, v U Proof. According to Lemma 3.6, the variety is commutative. If all nil-semi- ∈ ∈ V and u v, then u = v implies in the identity u = 0. Now it is very easy to groups in are singletons, then the desirable conclusion is evident. Suppose ≡ I V check that the lattice L( ) has the form shown on Fig. 1 where now that contains a non-singleton nil-semigroup N. Let be the vari- I V N = var x2yz = x x x =0,x2y = xy2, xy = yx where n 4, ety generated by N. Clearly, this variety is commutative. It is evident that In { 1 2 ··· n } ≥ . We need to verify that satisfies the identity (1.2). Put = var x2yz = x3 =0,x2y = xy2, xy = yx , ZM⊆N N J { } = var x2y = xy2, xy = yx, x2yz =0 = var x2yz = x3 = x x x =0,x2y = xy2, xy = yx where n 4, I { } Jn { 1 2 ··· n } ≥ 2 and = . It is clear that , whence . = var x y =0, xy = yx , N N ∧I ZM⊆I ZM⊆N K { } A semigroup analog of the proposition that we are verifying here is proved 2 n = var x y = x1x2 xn =0, xy = yx where n 3, in [22] (see the last paragraph of Section 3 in that article). The arguments used K { ··· } ≥ = var x2 =0, xy = yx , there are based on the fact that there is a variety such that the following L { } X 2 two claims are valid: n = var x = x1x2 xn =0, xy = yx where n N. L { ··· } ∈ (i) ( ) ; Note that = and = . X ∨N ∧V ⊆I L1 T L2 ZM 22 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers. I s p J p p In+1 s s p p K p n+1 n J I s s s p p p L p n p 4 n+1 J I s sK s s p p p Ln+1 p n p 4 s sK sJ p n p L K4 s s p 4 p L K3 s s 3 L s 2 L s 1 L s Figure 1. The lattice L( ) I Let . We have to check that if and is an arbitrary epigroup X ⊆I Y⊆X Z variety, then ( ) =( ) . For a variety with , we Z∨Y ∧X Z∧X ∨Y M M⊆I denote by ∗ the least of the varieties , , , and that contains . Fig. 1 M I J K L M shows that if , , then = if and only if deg( )=deg( ) M1 M2 ⊆I M1 M2 M1 M2 and ∗ = ∗. Therefore, we have to verify the following two equalities: M1 M2 deg ( ) = deg ( ) , (3.1) Z∨Y ∧X Z∧X ∨Y ( ) ∗ = ( ) ∗. (3.2) Z∨Y ∧X Z∧X ∨Y The equality (3.1). Put deg( )= k, deg( )=, and deg( )=m. Accord- X Y Z ing to Corollaries 2.13 and 2.14, we have deg ( ) = min max m, ,k , Z∨Y ∧X { } deg ( ) = max min m, k , . Z∧X ∨Y { } Clearly, k, because . It is then evident that ≤ Y⊆X if m k, ≤ ≤ min max m, ,k = max min m, k , = m if m k, { } { } ≤ ≤ k if k m. ≤ ≤ 22 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 23 The equality (3.1) is proved. I s The equality (3.2). Clearly, this equality is equivalent to the following p 3 2 2 p claim: if u is one of the words x , x y and x , then the variety ( ) J p n+1 Z∨Y ∧X I satisfies the identity u = 0 if and only if the variety ( ) does so. s s Z∧X ∨Y p It suffices to verify that u = 0 holds in ( ) whenever it is so in p K p n+1 n Z∨Y ∧X J I ( ) , because the opposite claim immediately follows from the evident s s s Z∧X ∨Y p inclusion ( ) ( ) . Further considerations are divided into p p Z∧X ∨Y ⊆ Z∨Y ∧X L p n p 4 two cases. n+1 J I s sK s s n n p Case 1: u x where n 2, 3 . Then x = 0 in ( ) . This means p p n ≡ ∈{ } Z∧X ∨Yn n+1 p p 4 that x = 0 in and there is a deduction of the identity x = 0 from the L n J Y s sK s identities of the varieties and . In particular, there is a word v such that Z X p v xn and xn = v holds in either or . If xn = v in , then the fact that n p 4 ≡ Z X X L s sK is a nilvariety together with the claims (i) and (ii) of Lemma 2.5 imply that Xn n p x = 0 in , and moreover in ( ) . Now let x = v in . The case 4 p X Z∨Y ∧X Z L K3 when v is a semigroup word may be considered by the same arguments as in s s the Case 1 in the ‘only if’ part of the proof of Theorem 2 in [27]. Now let v be 3 a non-semigroup word. Then, in view of Lemma 2.3, the identity v = 0 holds L s in the variety , and therefore in . Recall that xn = 0 in . Therefore, the X Y Y 2 identity xn = v holds in . Since this identity holds in as well, we obtain L Y Z s that it holds in . We see that the sequence xn, v, 0 is a deduction of n Y∨Z 1 the identity x = 0 from the identities of the varieties and , whence L Y∨Z X s xn = 0 holds in ( ) . Y∨Z ∧X Case 2: u x2y. We have to check that if the variety ( ) satisfies Figure 1. The lattice L( ) ≡ Z∧X ∨Y I the identity (1.1), then the variety ( ) also satisfies this identity. Put Z∨Y ∧X W = x2y, xyx, yx2,y2x, yxy, xy2 . The variety ( ) is commutative. { } Z∨Y ∧X Let . We have to check that if and is an arbitrary epigroup Therefore, it suffices to verify that this variety satisfies an identity w =0 X ⊆I Y⊆X Z for some word w W . By the hypothesis, the variety ( ) satisfies variety, then ( ) =( ) . For a variety with , we ∈ Z∧X ∨Y Z∨Y ∧X Z∧X ∨Y M M⊆I the identity (1.1). This means that this identity holds in and there is a denote by ∗ the least of the varieties , , , and that contains . Fig. 1 M I J K L M Y shows that if , , then = if and only if deg( )=deg( ) deduction of the identity from the identities of the varieties and . Let 1 2 1 2 1 2 2 X Z M M ⊆I M M M M x y u0,u1,...,un, 0 be such a deduction. The case when un W may be and 1∗ = 2∗. Therefore, we have to verify the following two equalities: ≡ ∈ M M considered by the same way as in the ‘only if’ part of the proof of Theorem 2 deg ( ) = deg ( ) , (3.1) Z∨Y ∧X Z∧X ∨Y in [27]. ( ) ∗ = ( ) ∗. (3.2) Now let un / W . Since u0 W , there is an index i>0 such that ui / W Z∨Y ∧X Z∧X ∨Y ∈ ∈ ∈ while ui 1 W . The identity ui 1 = ui holds in one of the varieties or . The equality (3.1). Put deg( )= k, deg( )=, and deg( )=m. Accord- − ∈ − Z X X Y Z If ui 1 = ui holds in , then satisfies the identity ui 1 = 0 (this follows ing to Corollaries 2.13 and 2.14, we have − X X − from [27, Lemma 2.5] whenever ui is a semigroup word and from Lemma 2.3, deg ( ) = min max m, ,k , otherwise). Therefore, ui 1 = 0 holds in ( ) . Since ui 1 W , we − Z∨Y ∧X − ∈ Z∨Y ∧X { } are done. deg ( ) = max min m, k , . Z∧X ∨Y { } Finally, suppose that ui 1 = ui holds in . If ui is a semigroup word, then − Z Clearly, k, because . It is then evident that we may complete the proof by the same arguments as in the ‘only if’ part ≤ Y⊆X of the proof of Theorem 2 in [27]. Now suppose that u is not a semigroup if m k, i ≤ ≤ word. Lemma 2.3 implies then that the variety satisfies the identity ui = 0. min max m, ,k = max min m, k , = m if m k, Y { } { } ≤ ≤ Hence, the identity ui 1 = ui holds in , and therefore the variety − Y Y∨Z k if k m. satisfies this identity. Applying Lemma 2.3 again, we conclude that u =0 ≤ ≤ i 24 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers. holds in . Whence, the sequence ui 1, ui, 0 is a deduction of the identity X − ui 1 = 0 from the identities of the varieties and . Thus, ui 1 =0 − Y∨Z X − holds in ( ) , and we are done. Y∨Z ∧X The equality (3.2) is proved. Thus, we have proved Proposition 3.8. Proof of Theorem 1.4. Necessity. Let be a strongly permutative upper- V modular epigroup variety. In view of Corollary 2.8, = where V G∨Cm ∨N is an abelian group variety, m 0, and is a nilvariety. Lemma 3.6 and G ≥ N Proposition 3.7 imply, respectively, that is commutative and satisfies the N identity (1.2). The variety contains a nilsubvariety var xm =0, xy = yx . Cm { } Clearly, this variety does not satisfy the identity (1.2) whenever m 3. Now ≥ Proposition 3.7 applies again and we conclude that m 2. If the variety ≤ N satisfies the identity (1.1), then the claim (ii) of Theorem 1.4 holds. Suppose now that the identity (1.1) fails in . By [29, Lemma 7], this implies that N N contains the variety . We need to verify that = and m 1. Arguing by J G T ≤ contradiction, suppose that either = or m 2. Then contains a variety G T ≥ V of the form where is either a non-trivial abelian group variety or the X ∨J X variety . The variety is generated by a 3-element monoid (see a remark C2 C2 before Lemma 2.7). Thus, is generated by an epigroup with unit, in any M case. Suppose that satisfies the identity (1.2). Substituting 1 for y in this X identity, we have that x2 = x holds in . But this identity is false both in X a non-trivial group variety and in the variety . As is verified in the proof C2 of [29, Lemma 8], this implies that (1.2) is false in any nil-semigroup in . X ∨J But this contradicts Proposition 3.7. Sufficiency. If satisfies the claim (i) of Theorem 1.4, then is upper- V V modular by Proposition 3.8 and Corollary 3.2. Suppose now that satisfies V the claim (ii). In other words, = where is an abelian group V G∨Cm ∨N G variety, 0 m 2, and satisfies the identities xy = yx and (1.1). Let ≤ ≤ N and an arbitrary epigroup variety. We aim to verify that Y⊆V Z ( ) =( ) . Z∨Y ∧V Z∧V ∨Y As we have already mentioned in the proof of Lemma 2.7, the variety Cm is generated by the (m + 1)-element combinatorial cyclic monoid Cm, and the set X = m N Cm has the greatest element. Indeed, for any m 0, { ∈ | ∈ X} ≥ let cm be a generator of Cm. Put C = m X Cm. Then the semigroup C is ∈ not an epigroup, because no power of the element (...,cm,...)m X belongs ∈ to a subgroup of C. Thus, the set X has the greatest element. We denote this element by m and put C( )= . It is clear that the varieties ( ) X Cm Z∨Y ∧V and ( ) are commutative. In view of Proposition 2.6 and Lemma 2.7, Z∧V ∨Y it suffices to verify the following three equalities: Gr ( ) = Gr ( ) , (3.3) Z∨Y ∧V Z∧V ∨Y C ( ) = C ( ) , (3.4) Z∨Y ∧V Z∧V ∨Y Nil ( ) = Nil ( ) . (3.5) Z∨Y ∧V Z∧V ∨Y 24 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 25 holds in . Whence, the sequence ui 1, ui, 0 is a deduction of the identity The equality (3.3). If is a periodic group variety, then we denote by X − G ui 1 = 0 from the identities of the varieties and . Thus, ui 1 =0 exp( ) the exponent of , that is, the least number n such that satisfies − Y∨Z X − G G G holds in ( ) , and we are done. the identity x = xn+1. For a non-periodic group variety , we put exp( )= Y∨Z ∧X G G The equality (3.2) is proved. Thus, we have proved Proposition 3.8. . As usual, we denote by lcm m, n (respectively, gcd m, n ) the least ∞ { } { } common multiple (the greatest common divisor) of positive integers m and n. Proof of Theorem 1.4. Necessity. Let be a strongly permutative upper- V To simplify further considerations, we will assume that any natural number modular epigroup variety. In view of Corollary 2.8, = where V G∨Cm ∨N divides ; in particular, gcd n, = n and lcm n, = for arbitrary is an abelian group variety, m 0, and is a nilvariety. Lemma 3.6 and ∞ { ∞} { ∞} ∞ G ≥ N natural n. We will assume also that gcd , = lcm , = . Put Proposition 3.7 imply, respectively, that is commutative and satisfies the {∞ ∞} {∞ ∞} ∞ N m 1 = Gr ( ) and 2 = Gr ( ) . Since 1, 2 , we have identity (1.2). The variety m contains a nilsubvariety var x =0, xy = yx . G Z∨Y ∧V G Z∧V ∨Y G G ⊆V C { } that 1 and 2 are abelian group varieties. To prove the equality (3.3), it Clearly, this variety does not satisfy the identity (1.2) whenever m 3. Now G G ≥ suffices to verify that exp( 1) = exp( 2). This claim is verified by the same Proposition 3.7 applies again and we conclude that m 2. If the variety G G ≤ N arguments as the analogous claim in the proof of Theorem 1.2 in [22]. satisfies the identity (1.1), then the claim (ii) of Theorem 1.4 holds. Suppose The equality (3.4). Here and below we need the following easy remark. It now that the identity (1.1) fails in . By [29, Lemma 7], this implies that N N is evident that if m 3, then the identity (1.1) fails in the variety Nil( m)= contains the variety . We need to verify that = and m 1. Arguing by ≥ C J G T ≤ var xm =0, xy = yx . This means that each part of the equality (3.4) coin- contradiction, suppose that either = or m 2. Then contains a variety { } G T ≥ V cides with the variety m for some 0 m 2. Then we may complete the of the form where is either a non-trivial abelian group variety or the C ≤ ≤ X ∨J X proof of equality (3.4) by the same arguments as in the proof of the equal- variety . The variety is generated by a 3-element monoid (see a remark C2 C2 ity (4.2) in [22]. before Lemma 2.7). Thus, is generated by an epigroup with unit, in any 2 M The equality (3.5). The varieties , 2, and satisfy the identity x y = case. Suppose that satisfies the identity (1.2). Substituting 1 for y in this G C N X x 2y. By Lemma 2.3, the variety Nil( ) satisfies the identity (1.1). Fig. 1 identity, we have that x2 = x holds in . But this identity is false both in V X shows that it suffices to check the following two claims: first, the varieties a non-trivial group variety and in the variety . As is verified in the proof C2 ( ) and ( ) have the same degree, and second, the variety of [29, Lemma 8], this implies that (1.2) is false in any nil-semigroup in . Z∨Y ∧V Z∧V ∨Y X ∨J Nil ( ) satisfies the identity But this contradicts Proposition 3.7. Z∨Y ∧V Sufficiency. If satisfies the claim (i) of Theorem 1.4, then is upper- V V 2 modular by Proposition 3.8 and Corollary 3.2. Suppose now that satisfies x = 0 (3.6) V the claim (ii). In other words, = where is an abelian group V G∨Cm ∨N G variety, 0 m 2, and satisfies the identities xy = yx and (1.1). Let if and only if the variety Nil ( ) ) satisfies this identity. The former ≤ ≤ N Z∧V ∨Y and an arbitrary epigroup variety. We aim to verify that claim may be verified by the same way as in the proof of the equality (4.3) Y⊆V Z ( ) =( ) . in [22] with references to Proposition 2.12, Corollary 2.14, and Corollary 2.15 of Z∨Y ∧V Z∧V ∨Y the present article rather than Proposition 2.11, Lemma 2.13, and Lemma 2.12 As we have already mentioned in the proof of Lemma 2.7, the variety Cm of [22], respectively. is generated by the (m + 1)-element combinatorial cyclic monoid Cm, and the It remains to verify that the variety Nil ( ) satisfies the iden- Z∨Y ∧V set X = m N Cm has the greatest element. Indeed, for any m 0, tity (3.6) whenever Nil ( ) does (because the opposite claim is ev- { ∈ | ∈ X} ≥ Z∧V ∨Y let cm be a generator of Cm. Put C = m X Cm. Then the semigroup C is ident). Suppose that the identity (3.6) holds in Nil ( ) . Propo- ∈ Z∧V ∨Y not an epigroup, because no power of the element (...,cm,...)m X belongs sition 2.6 and Lemma 2.7 imply that the variety ( ) is the join ∈ Z∧V ∨Y to a subgroup of C. Thus, the set X has the greatest element. We denote this of some group variety, the variety m for some m 0, and the variety element by m and put C( )= . It is clear that the varieties ( ) C ≥ X Cm Z∨Y ∧V Nil ( ) . Here m 2, because ( ) . Hence, the vari- and ( ) are commutative. In view of Proposition 2.6 and Lemma 2.7, Z∧V ∨Y ≤ Z∧V ∨Y ⊆V Z∧V ∨Y ety ( ) satisfies the identity x2 = x2. In particular, this identity it suffices to verify the following three equalities: Z∧V ∨Y holds in the varieties and . Therefore, there is a sequence of words Y Z∧V Gr ( ) = Gr ( ) , (3.3) u ,u ,...,u such that u x2, u x2, and for each i =0, 1,...,k 1, Z∨Y ∧V Z∧V ∨Y 0 1 k 0 ≡ k ≡ − the identity u = u holds in one of the varieties or . We may assume C ( ) = C ( ) , (3.4) i i+1 Z V Z∨Y ∧V Z∧V ∨Y that ui ui+1 for each i =0, 1,...,k 1. Arguments from the proof of the Nil ( ) = Nil ( ) . (3.5) ≡ − Z∨Y ∧V Z∧V ∨Y equality (4.3) in [22] show that it suffices to check that the identity (3.6) holds 26 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers. in one of the varieties Nil( ) or Nil( ). This fact follows from Lemma 2.5 Z V whenever u1 is a semigroup word and from Lemma 2.3 otherwise. We have completed the proof of Theorem 1.4. 4. Corollaries Comparing Proposition 4.1 in [31] with Theorem 1.1 leads to the following fact. Corollary 4.1. Let be a periodic semigroup variety. Then is a neutral V V element of SEM if and only if is a neutral element of Epi. V Analogously, Theorem 3.1 in [19] and Theorem 1.3 show that the following is true. Corollary 4.2. Let be a periodic commutative semigroup variety. Then V V is a modular element of SEM if and only if is a modular element of Epi. V Our next corollary is not so immediate a consequence of earlier results as the previous two statements. Corollary 4.3. Let be a periodic strongly permutative semigroup variety. V Then is an upper-modular element of SEM if and only if is an upper- V V modular element of Epi. Proof. Necessity. Let be a periodic strongly permutative semigroup variety V and is an upper-modular element of SEM. It follows from results of [5] V and the proof of Proposition 1 in [29] that = for some Abelian V G∨Cm ∨N periodic group variety , some m 0 and some nilvariety . By [22, Theo- V ≥ N rem 1.1], if is a proper semigroup variety that is an upper-modular element X of SEM and is a nilsubvariety of then is commutative. Thus, is Y X Y N commutative. This implies that the variety is commutative too. Now we V may apply [22, Theorem 1.2] and conclude that satisfies one of the claims (i) V or (ii) of Theorem 1.4. Therefore, is upper-modular in Epi. V Sufficiency. Let be a periodic strongly permutative semigroup variety V and an upper-modular element of Epi. By Theorem 1.4, either satisfies V V the claim (i) of this theorem or satisfies the claim (ii) of this theorem and V the variety from this claim is periodic. In both these cases, is an upper- G V modular element of SEM by [22, Theorem 1.2]. Theorems 1.2 and 1.4 evidently imply the following assertion. Corollary 4.4. If a commutative epigroup variety is a modular element of the lattice Epi, then it is an upper-modular element of this lattice. Theorem 1.4 readily implies the following statement. Corollary 4.5. If a strongly permutative epigroup variety is an upper- V modular element of the lattice Epi, then the lattice L( ) is distributive. V 26 V. Yu. Shaprynskiˇı, D. V. Skokov, and B. M. Vernikov Algebra univers. Vol. 00, XX The latticelattice ofof epigroupepigroup varietiesvarieties 27 in one of the varieties Nil( ) or Nil( ). This fact follows from Lemma 2.5 Proof. If satisfies the claim (i) of Theorem 1.4, then it is periodic, whence it Z V V whenever u1 is a semigroup word and from Lemma 2.3 otherwise. may be considered as a semigroup variety. In this case, it suffices to take into We have completed the proof of Theorem 1.4. account a description of commutative semigroup varieties with distributive subvariety lattice obtained in [30]. Suppose now that satisfies the claim (ii) V 4. Corollaries of Theorem 1.4. Then where satisfies the commutative V ⊆ AG ∨ C2 ∨N N law and the identity (1.1). In view of Proposition 2.9, L( ) is embeddable into V Comparing Proposition 4.1 in [31] with Theorem 1.1 leads to the following the direct product of the lattices L( ) and L( ). The former lattice AG C2 ∨N fact. is generally known to be distributive. Finally, the variety is periodic, C2 ∨N whence it may be considered as a semigroup variety. To complete the proof, Corollary 4.1. Let be a periodic semigroup variety. Then is a neutral V V it remains to note that the lattice L( ) is distributive by the mentioned element of SEM if and only if is a neutral element of Epi. C2 ∨N V result of [30]. Analogously, Theorem 3.1 in [19] and Theorem 1.3 show that the following is true. 5. Open questions Corollary 4.2. Let be a periodic commutative semigroup variety. Then V V We do not know whether it is possible to remove the word ‘commutative’ is a modular element of SEM if and only if is a modular element of Epi. V in Corollary 4.2 and the words ‘strongly permutative’ in Corollary 4.3. One Our next corollary is not so immediate a consequence of earlier results as can formulate the corresponding questions. the previous two statements. Question 5.1. Is it true that a periodic semigroup variety is a modular ele- Corollary 4.3. Let be a periodic strongly permutative semigroup variety. ment of SEM if and only if it is a modular element of Epi? V Then is an upper-modular element of SEM if and only if is an upper- V V Question 5.2. Is it true that a periodic semigroup variety is an upper-modular modular element of Epi. element of SEM if and only if it is an upper-modular element of Epi? Proof. Necessity. Let be a periodic strongly permutative semigroup variety V It was proved in [21, Corollary 3.5] that the following strengthened semi- and is an upper-modular element of SEM. It follows from results of [5] V group analog of Theorem 1.1 is true: a semigroup variety is neutral in SEM and the proof of Proposition 1 in [29] that = for some Abelian V G∨Cm ∨N if and only if it is simultaneously lower-modular and upper-modular in SEM. periodic group variety , some m 0 and some nilvariety . By [22, Theo- V ≥ N We do not know, whether the epigroup analog of this claim is valid. rem 1.1], if is a proper semigroup variety that is an upper-modular element X of SEM and is a nilsubvariety of then is commutative. Thus, is Question 5.3. Is it true that an epigroup variety is a neutral element of Y X Y N commutative. This implies that the variety is commutative too. Now we the lattice Epi if and only if it is simultaneously a lower-modular and upper- V may apply [22, Theorem 1.2] and conclude that satisfies one of the claims (i) modular element of this lattice? V or (ii) of Theorem 1.4. Therefore, is upper-modular in Epi. As we have already mentioned in Subsection 3.5, it is proved in [22, The- V Sufficiency. Let be a periodic strongly permutative semigroup variety orem 1.1] that if is a proper upper-modular in SEM semigroup variety, V V and an upper-modular element of Epi. By Theorem 1.4, either satisfies then every nilsubvariety of is commutative and satisfies the identity (1.2). V V V the claim (i) of this theorem or satisfies the claim (ii) of this theorem and Proposition 3.7 gives a partial epigroup analog of this assertion. We do not V the variety from this claim is periodic. In both these cases, is an upper- know whether the full analog is true. G V modular element of SEM by [22, Theorem 1.2]. Question 5.4. Suppose that an epigroup variety is an upper-modular ele- V Theorems 1.2 and 1.4 evidently imply the following assertion. ment of the lattice Epi and let be a nilsubvariety of . N V (a) Is it true that is commutative? Corollary 4.4. If a commutative epigroup variety is a modular element of the N (b) Is it true that satisfies the identity (1.2)? lattice Epi, then it is an upper-modular element of this lattice. N Proposition 3.7 shows that an affirmative answer to Question 5.4(a) would Theorem 1.4 readily implies the following statement. immediately imply the same answer to Question 5.4(b). Corollary 4.5. If a strongly permutative epigroup variety is an upper- Further, it is verified in [23, Theorem 1] that every proper upper-modular V modular element of the lattice Epi, then the lattice L( ) is distributive. in SEM variety is either commutative or has a degree 2. We do not know V ≤ 28 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers. whether the epigroup analog of this alternative is valid. We formulate the corresponding question together with its weaker version. Question 5.5. Suppose that an epigroup variety is an upper-modular ele- V ment of the lattice Epi. (a) Is it true that either is commutative or has a degree 2? V ≤ (b) Is it true that either is permutative or has a finite degree? V An affirmative answer to Question 5.5(a) together with Theorem 1.4 would immediately imply a complete description of upper-modular epigroup varieties of degree > 2. At the conclusion of the article, we briefly mention lower-modular and dis- tributive varieties. It is verified in [20] that every proper lower-modular in SEM variety is periodic (this fact is essentially generalized by [12, Theo- rem 1]). The question, whether the epigroup analog of this claim is the case, is open. Question 5.6. Is it true that if an epigroup variety is a lower-modular V element of the lattice Epi, then the variety is periodic? V Lower-modular in SEM varieties were completely determined in [13] (and this result is reproved in a simpler way in [11]). Further, it is evident that a distributive element of a lattice is lower-modular.Varieties distributive in SEM were completely classified in [25]. This allows us to hope that the affirmative answer to Question 5.6 would result in a complete description of the lower- modular and the distributive epigroup varieties. Acknowledgments. The authors thank anonymous referee for a long list of useful suggestions on improvement of the initial version of the manuscript. References [1] Golubov, E.A., Sapir, M.V.: Residually small varieties of semigroups. Izv. VUZ Matem. No. 11, 21–29 (1982) (Russian; Engl. translation: Soviet Math. (Iz. VUZ) 26, No. 11, 25–36 (1982)) [2] Gr¨atzer,G.: Lattice Theory: Foundation. Birkh¨auser,Springer Basel AG (2011) [3] Gr¨atzer,G., Schmidt, E.T.: Standard ideals in lattices. Acta Math. Acad. Sci. Hungar. 12, 17–86 (1961) [4] Gusev, S.V., Vernikov, B.M.: Endomorphisms of the lattice of epigroup varieties. Semigroup Forum (in press); available at http:// arxiv.org/abs/1404.0478v4 [5] Head, T.J.: The lattice of varieties of commutative monoids. Nieuw Arch. Wiskunde. Ser. III. 16, 203–206 (1968) [6] JeˇzekJ., McKenzie, R.N.: Definability in the lattice of equational theories of semigroups. Semigroup Forum 46, 199–245 (1993) [7] Petrich, M.: Inverse Semigroups. 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[28] Vernikov, B.M., Volkov, M.V., Shaprynskiˇı,V.Yu.: Semimodular and Arguesian Semigroup Forum (in press); available at http:// arxiv.org/abs/1404.0478v4 varieties of epigroups. II (Russian) (in press) [5] Head, T.J.: The lattice of varieties of commutative monoids. Nieuw Arch. Wiskunde. [29] Volkov, M.V.: Semigroup varieties with modular subvariety lattices. Izv. VUZ Matem. Ser. III. 16, 203–206 (1968) No. 6, 51–60 (1989) (Russian; Engl. translation: Soviet Math. (Iz. VUZ) 33, No. 6, [6] JeˇzekJ., McKenzie, R.N.: Definability in the lattice of equational theories of 48–58 (1989)) semigroups. Semigroup Forum 46, 199–245 (1993) [30] Volkov, M.V.: Commutative semigroup varieties with distributive subvariety lattices. [7] Petrich, M.: Inverse Semigroups. Wiley Interscience, New York (1984) Contrib. General Algebra 7, 351–359 (1991) [8] Petrich, M., Reilly, N.R.: Completely Regular Semigroups. John Wiley & Sons, New [31] Volkov, M.V.: Modular elements of the lattice of semigroup varieties. Contrib. York (1999) General Algebra 16, 275–288 (2005) [9] Sapir, M.V., Sukhanov, E.V.: On varieties of periodic semigroups. Izv. VUZ Matem. No. 4, 48–55 (1981) (Russian; Engl. translation: Soviet Math. (Iz. VUZ) 25, No. 4, 53–63 (1981)) 30 V. Yu. Shaprynskiˇı, Shaprynskiˇı, D. D. V. V. Skokov, Skokov, and and B. M. Vernikov AlgebraAlgebra Univers. univers. Vyacheslav Yu. Shaprynskiˇı Institute of Mathematics and Computer Science, Ural Federal University, Lenina str. 51, 620000 Ekaterinburg, Russia e-mail: [email protected] Dmitry V. Skokov Institute of Mathematics and Computer Science, Ural Federal University, Lenina str. 51, 620000 Ekaterinburg, Russia e-mail: [email protected] Boris M. Vernikov Institute of Mathematics and Computer Science, Ural Federal University, Lenina str. 51, 620000 Ekaterinburg, Russia e-mail: [email protected]