We prove that the congruence lattice of a nilsemigroup is modular if and only if the width of the semigroup, as a poset, is at most two, and distributive if and only if its width is one. In the latter case, such semigroups therefore coincide with the nil Δ \u3eΔ Δ -semigroups. It is further shown that if a finitely generated nilsemigroup has modular congruence lattice, then the semigroup is finite

Jones, Peter

Popovich, Alexander L.

Semigroup Forum

English

Alexander L. Popovich

Peter R. Jones

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On congruence lattices of nilsemigroups

We prove that the congruence lattice of a nilsemigroup is modular if and only if the width of the semigroup, as a poset, is at most two, and distributive if and only if its width is one. In the latter case, such semigroups therefore coincide with the nil Δ -semigroups. It is further shown that if a finitely generated nilsemigroup has modular congruence lattice, then the semigroup is finite. © 2016, Springer Science+Business Media New York

Popovich, A. L.

Jones, P. R.

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Marquette Universitye-Publications@MarquetteMathematics, Statistics and Computer ScienceFaculty Research and PublicationsMathematics, Statistics and Computer Science,Department of10-1-2017On Congruence Lattices of NilsemigroupsAlexander L. PopovichUral Federal UniversityPeter JonesMarquette University, peter.jones@marquette.eduAccepted version. Semigroup Forum, Vol. 95, No. 2 (October 2017): 314-320. DOI. © 2017 SpringerInternational Publishing AG. Part of Springer Nature. Used with permission.Shareable Link. Provided by the Springer Nature SharedIt content-sharing initiative.On congruence lattices of nilsemigroupsAlexander L. Popovich∗ and Peter R. JonesAbstractWe prove that the congruence lattice of a nilsemigroup is modular if and only if thewidth of the semigroup, as a poset, is at most two, and distributive if and only if its widthis one. In the latter case, such semigroups therefore coincide with the nil ∆-semigroups. Itis further shown that if a finitely generated nilsemigroup has modular congruence lattice,then the semigroup is finite.Studying congruence lattices is a well-established direction in semigroup theory. Surveys ofthis area have been given in [13] and [14]. One of the natural questions here is to characterizesemigroups with distributive or modular congruence lattices. This problem has been solved forseveral classes of semigroups.Of course the congruence lattice of every group is modular. The characterization of abeliangroups whose congruence lattices are distributive is a corollary of the famous result of Ore [16].Non-abelian groups with distributive congruence lattices had been studied by Pazderski [17] andMaj [12]. The case of semilattices had been considered by Dean and Oehmke [4] and Hamilton[8], who showed that the congruence lattice of a semilattice is modular if and only if it isdistributive, and if and only if the semilattice itself is a tree. Fountain and Lockley [5] classifiedthe Clifford semigroups with either modular or distributive congruence lattice and the sameauthors [6] determined the bands with distributive congruence lattice. Auinger [1] studied strictinverse semigroups with distributive or modular congruence lattices. Bonzini and Cherubini [2]studied the case of inverse ω-semigroups. Regular semigroups with the minimal condition foridempotents and having distributive or modular congruence lattices were characterized by Jones[10]. Hamilton [9] studied the case of commutative cancellative semigroups. The ∆-semigroups– semigroups whose congruence lattices form a chain – have been particularly well studied (see,for instance, [15]).In the present paper we are interested in the class of nilsemigroups. Recall that a semigroupwith zero is called a nilsemigroup if, for each of its elements x, there exists a positive integern such that xn = 0. A little is already known about congruence lattices of nilsemigroups. Itfollows from [11] that the congruence lattice L of a nilsemigroup is strictly semimodular, i.e. itCommunicated by Mikhail V. Volkov∗The first author acknowledges support from the Presidential Programme “Leading Scientific Schools of theRussian Federation”, project no. 5161.2014.1, the Russian Foundation for Basic Research, project no. 14-01-00524, the Ministry of Education and Science of the Russian Federation, project no. 1.1999.2014/K, and theCompetitiveness Program of Ural Federal University.1satisfies the following property: for every a, b, c ∈ L, a  b implies a∨ c  b∨ c or a∨ c = b∨ c,where  is the covering relation (see [7]). As is noted below, it is easy to see that the class ofcongruence lattices of nilsemigroups satisfies no nontrivial lattice identity.Other than in the study of ∆-semigroups, until now nothing has been known in general aboutnilsemigroups with distributive or modular congruence lattices. We reduce the distributive caseto that of ∆-semigroups and then characterize the modular case in terms of the width of theunderlying poset of the semigroup.Let S be a nilsemigroup. It is a well-known fact that S is J -trivial. Thus the naturalpartial order on the J -classes reduces to a partial order on S itself, given byy 6 x iff there exist s, t ∈ S1 such that y = sxt.Let (X,6) be any poset. The notation a ‖ b indicates that a and b are incomparable under6. A subset A ⊆ X is called an antichain [a chain] if elements of A are pairwise incomparable[pairwise comparable] under 6. The width of (X,6) is the greatest cardinality, if it exists, ofany antichain in X.For the elementary background on semigroups and lattices needed here we refer the readerto [3] and [7] respectively. A lattice (L,∧,∨) is distributive if a∧ (b∨ c) = (a∧ b)∨ (a∧ c) for alla, b, c ∈ L. Among several characterizations of modularity of L is the following: if a 6 c, then(a ∨ b) ∧ c = a ∨ (b ∧ c), for all a, b, c ∈ L.Denote the congruence lattice of a semigroup S by (ConS, ∩ ,∨), or just ConS.The main results of the paper are the following:Theorem 1. Let S be a nilsemigroup. Then the following are equivalent:1) ConS is distributive;2) The poset (S,6) is a chain;3) ConS is a chain.Theorem 2. Let S be a nilsemigroup. Then the following are equivalent:1) ConS is modular but not distributive;2) The poset (S,6) has width 2.According to Theorem 1, a nilsemigroup has distributive congruence lattice if and only ifit is a ∆-semigroup. In the finite case, it follows from earlier work on such semigroups thatthe semigroup must be cyclic. Following Proposition 2, we include the proof of a slightly moregeneral statement, for completeness.In the infinite case, the commutative ∆-semigroups were completely described by Tamura[21] and Schein [19, 20]. Specializing to the case of nilsemigroups, the following completedescription is obtained. Here the semigroups Q and R are the Rees quotients of the semigroupR+ of positive real numbers, under addition, modulo the ideals [1,∞) and (1,∞), respectively.Corollary 1. An infinite, commutative nilsemigroup has distributive congruence lattice if andonly if it is isomorphic to a subsemigroup G of Q or of R with the property that whenever x ∈ Gand x+ y ∈ G\0, then y ∈ G.2Corollary 2. Let S be a nilsemigroup such that ConS is modular. If S is finitely generated,then it is finite. If S is not cyclic, then it is generated by two elements a and b, say, and theposet {a2, ab, ba, b2} has width at most two.Examples of infinite nilsemigroups with congruence lattices that are modular but not dis-tributive are easily constructed. For instance, the 0-direct union of any two infinite, totallyordered, nilsemigroups is a nilsemigroup of width two. Corollary 1 provides a wealth of candi-dates from which to build.The following elementary fact is a part of semigroup folklore.Lemma 1. Let S be a nilsemigroup, a, b ∈ S and a 6= 0. Then a > ab and a > ba.Lemma 2. Let S be a nilsemigroup, θ ∈ ConS, (a, b) ∈ θ and a > b. Then (a, 0) ∈ θ.Proof. Since a > b, then b = sat for some s, t ∈ S1 and either s 6= 1 or t 6= 1. By assumption,there exists n such that sn = 0 or tn = 0, respectively. Then(b, sbt) = (sat, sbt) ∈ θ, (sbt, s2bt2) ∈ θ, . . . , (sn−1btn−1, 0) ∈ θ.By transitivity, (a, 0) ∈ θ.Denote by Partn the full partition lattice of a set with n elements. It is well known thatPartn is non-distributive for n > 3 and non-modular for n > 4. Moreover, the class of all suchfinite partition lattices satisfies no proper lattice identity [18]. Observe that the congruencelattice of any n-element null (or ‘zero’) semigroup is Partn itself. Since such semigroups arenilsemigroups, it follows that the class of congruence lattices of finite nilsemigroups satisfies nonontrivial lattice identity.Proposition 1. Let S be a nilsemigroup for which (S,6) contains an antichain of size n. ThenConS has a sublattice isomorphic to Partn+1.Proof. Let A be an n-element antichain in (S,6) andK = {x ∈ S | x < a for some a ∈ A}.It follows from Lemma 1 that K is an ideal of S. Consider the Rees quotient T = S/K. Theset I = A ∪ {0} forms an (n + 1)-element ideal in T . Let pi be a partition of I. If (x, y) ∈ pi,then for every t ∈ T , tx = xt = yt = ty = 0. Therefore the congruence generated by pi hasthe form pi ∪ ∆T , where ∆T denotes the equality relation on T . Thus we have the mappingf : Partn+1 → ConT defined by pi 7→ pi ∪∆T . It is easy to verify that f is a lattice embedding.Therefore Partn+1 is isomorphic to a sublattice of ConT , which in turn is isomorphic to a filterof ConS. So Partn+1 is isomorphic to a sublattice of ConS.In view of the properties of partition lattices cited earlier, that (1) implies (2) in Theorems 1and 2 is immediate. We now turn to the converses. In the case of distributivity, this is alreadyincluded in the proof of [15, Theorem 1.56]. Since it is easy, we include a proof for completeness.3Proposition 2. Let S be a nilsemigroup such that (S,6) is a chain. Then the ideals of S aretotally ordered and every congruence is a Rees ideal congruence. Hence ConS is a chain.Proof. That the ideals are totally ordered is clear. Now let θ ∈ ConS and denote by I theθ-class of 0. Clearly I is an ideal of S. If a ∈ S and the θ-class of a is not a singleton, then byLemma 2, a ∈ I. That is, θ is the Rees ideal congruence modulo I.In the introduction, it was noted that every finite nilsemigroup, whose congruence latticeis a chain, is cyclic. Again, we include the proof for completeness and for comparison with themodular case. In fact we show that any finitely generated nilsemigroup S such that (S,6) is achain must be finite cyclic. For suppose that x1 > x2 > · · · > xk is an irredundant generatingset for S, with k > 1. Since x2 < x1, x2 = sx1t for some s, t ∈ S1, not both 1. But then sand t, if not 1, must be power of x1 and so the same is true of x2 itself, contradicting theassumption. Thus S is cyclic. But if S is infinite, it has the finite cyclic groups as quotientsand therefore their congruence lattices as filters in ConS. Hence S is finite cyclic. (Finitenessis also a consequence of Corollary 2.)The converse argument in the case of modularity is somewhat more complex.Proposition 3. The congruence lattice of any nilsemigroup S such that (S,6) has width twois modular.Proof. Let ρ, θ, τ be congruences on S such that ρ ⊆ τ . Let (a, b) ∈ (ρ ∨ θ) ∩ τ . It must beshown that (a, b) ∈ ρ ∨ (θ ∩ τ).There is a sequence a = x0, x1, . . . , xk, xk+1 = b such that (xi, xi+1) ∈ ρ ∪ θ for each i. Wemay proceed by induction on k, the case k = 1 being obvious. In fact, since ρ ⊆ τ , it may beassumed that a θ x1 and xk θ b, so that k ≥ 2. It may, further, be assumed that each pair(xi, xi+1) belongs to exactly one of ρ and θ.First suppose one of a and b is zero, say b = 0. It follows from Lemma 2 that the θ-class ofxk is an ideal of S. Observe next that if xi < a for any i, 1 6 i 6 k, then (by the same lemma)a τ xi τ 0 and the induction hypothesis applies. Now consider xk−1 ρ xk. If these elementsare comparable, then xk−1 ρ 0 and the induction hypothesis applies. Otherwise, by the widthhypothesis, a is comparable with, and thus necessarily less than, one of the two. If a < xk, thena θ 0 (and so (a, 0) ∈ θ ∩ τ). So a < xk−1. Write a = sxk−1t for some s, t ∈ S1, not both 1.(We shall omit mention of this qualification in similar situations below and in the next proof.)Then a ρ sxkt θ 0 provides a shorter sequence and the induction hypothesis again applies.In the general case, suppose a and b are comparable. By Lemma 2, (a, 0), (b, 0) ∈ (ρ∨ θ)∩ τand so the previous case completes the argument.Otherwise a ‖ b. Suppose a and x1 are comparable and, also, that xk and b are comparable.Then a θ 0 θ b and so (a, b) ∈ θ ∩ τ . Without loss of generality, it may therefore be assumedthat a ‖ x1 and, by the width hypothesis, x1 and b are therefore comparable.Suppose that x1 > b. Note that since (x1, b) ∈ ρ∨ θ, then by Lemma 2, (b, 0) ∈ ρ∨ θ. Thusif it should happen that a τ 0 or b τ 0, then the proof of that special case applies. If x1 and x24are comparable, then x1 ρ 0 and so x1 ρ b, resulting in a shorter sequence. So x1 ‖ x2 and, bythe width hypothesis, x2 and a are comparable.If x2 < a, then, since we may write b = sx1t, we have b ρ sx2t < x2 < a. From a τ b it thenfollows that a τ 0.Alternatively, x2 > a and we may write a = qx2r ρ qx1r < x1. Since a ‖ x1, a 6= qx1r. Thusif a and qx1r are comparable, then by Lemma 2, a ρ 0 and so a τ 0. Otherwise, a ‖ qx1r and,by the width hypothesis, qx1r and b are comparable. If they are distinct, then b τ a τ qx1rimplies b τ 0. If b = qx1r, then a ρ b. This concludes the analysis in the case that x1 > b.Now suppose x1 < b, x1 = sbt, say, and consider xk. If xk and b are comparable, then b θ 0and so b θ x1 θ a. So we may assume xk ‖ b. Then after applying the analysis for the casex1 > b to the case xk > a, it remains to consider xk < a. Now a θ x1 = sbt θ sxkt < xk < a.Thus a θ 0 and a θ xk θ b. This completes the proof.Now we show how Theorems 1 and 2 imply Corollary 2.Proof. Since S is finitely generated, then it has an irredundant generating set B = {a1, . . . , an}.Suppose ai < aj for some i, j. Then there exist s, t such that ai = sajt. By irredundancy, anyexpression for sajt as a product from B must involve ai, which by Lemma 1 implies ai = 0,contradicting irredundancy. Thus B is an antichain under 6. By Theorems 1 and 2, |B| = 1 or|B| = 2. If |B| = 1, then S is a cyclic nilsemigroup, which is finite. Let |B| = 2 and B = {a, b},for convenience of notation. Then every element of S can be written as ak0bl1ak1bl2 . . . aknbln+1for some ki, li > 0 for 1 6 i 6 n, and k0, ln+1 > 0.Suppose that ab = ba. Then every element of S can be represented as akbl for suitable k, l.Since an = bm = 0 for some n,m, then S can contain only finitely many elements.Otherwise, without loss of generality we assume ab 6 ba. Thus ab 6= 0. Let a2 > ab. Thenthere exist s, t such that ab = sa2t. By Lemma 1, ab = ap for some p. Then every element ofS can be written as b`ak for suitable k, `, which, as before, means that S is finite. The caseb2 > ab is similar.Now let a2 6> ab, b2 6> ab. Suppose that ba < ab. Then ba = sabt. If s and t (if not 1) arenot respectively powers of a and of b, then by Lemma 1, ba = 0. In either event, we can writeevery element of S in the form akb`, so S is finite.Finally, let ab ‖ ba. We can assume that a2 6> ba and b2 6> ba (if not, then we have thesame arguments for ba as we had before for ab). Since S has width two, then a2 6 c and b2 6 dfor c, d ∈ {ab, ba}. Without loss of generality we may suppose that a2 < ab and a2 6< b2. Noweither a2 = 0 or from a2 = sabt it then follows from Lemma 1 that a2 = aibaba . . . baj for somei, j ∈ {0, 1}.Similarly, b2 = uabv or b2 = ubav and either b2 = 0 or, using the alternatives for a2 juststated, b2 = afbaba . . . bag for some f, g ∈ {0, 1}. Therefore, in all cases every element of S canbe written in the latter form and, since (ab)n = 0 for some n, S is finite.The authors are grateful to Vladimir Repnitskii, for his attention to the paper, and to thereferee for suggestions that improved its exposition.5References[1] Auinger, K.: The congruence lattice of a strict regular semigroup, J. Pure Appl. Algebra81 (1992), 219–245.[2] Bonzini, C., Cherubini, A.: Modularity of the lattice of congruences of a regular ω-semigroup, Proc. Edinburgh Math. Soc. 33 (1990), 405–407.[3] Clifford, A.H., Preston, G.B.: The algebraic theory of semigroups. Vol. I and II, Math-ematical Surveys, Number 7, American Mathematical Society, Providence, Rhode Island(1961, 1967).[4] Dean, R.A., Oehmke, R.H.: Idempotent semigroups with distributive right congruencelattices, Pacific J. Math. 14 (1964), 1187–1209.[5] Fountain, J.B., Lockley, P., Semilattices of groups with distributive congruence lattice,Semigroup Forum 14 (1977), 81–92.[6] Fountain, J.B., Lockley, P., Bands with distributive congruence lattice, Proc. Royal Soc.Edinb. (Sec. A) 84 (1979), 235–247.[7] Gra¨tzer, G.: Lattice Theory: Foundation, Birkha¨user Verlag, Basel (2011).[8] Hamilton, H.B.: Semilattices whose structure lattice is distributive, Semigroup Forum 8(1974), 245–253.[9] Hamilton, H.: Modularity and distributivity of the congruence lattice of a commutativeseparative semigroup, Math. Japan. 27 (1982), 581–589.[10] Jones, P.: On the congruence lattices of regular semigroups, J. Algebra 82 (1983), 18–39.[11] Jones, P.: Congruence semimodular varieties of semigroups, Lecture Notes Math. 1320(1988), 162–171, in Semigroups, theory and applications (Oberwolfach, 1986), Springer,Berlin.[12] Maj, M.: Gruppi infiniti supersolubili con il reticulo dei sottogruppi normali distributive,Rend. Accad. Sci. Fis. Math. Napoli 51 (1984), 15–20.[13] Mitsch, H.: Semigroups and their lattice of congruences. Semigroup Forum 26 (1983),1–63.[14] Mitsch, H.: Semigroups and their lattice of congruences II, Semigroup Forum 54 (1997),1–42.[15] Nagy, A., Special Classes of Semigroups, Kluwer Academic Publishers, Dor-drecht/Boston/London, 2001.[16] Ore, O.: Structures and group theory. Duke Math. J., 21 (1938), 247–269.6[17] Pazderski, G.: On groups for which the lattice of normal subgroups is distributive. Beitra¨geAlgebra Geom. 24 (1987), 185–200.[18] Sachs, D., Identities in finite partition lattices, Proc. Amer. Math. Soc. 12 (1961), 944–945.[19] Schein, B.M.: Commutative semigroups where congruences form a chain, Bull. Acad.Polon. Sci. Ser. Sci. Math. Astronom. Phys. 17 (1969), 523–527.[20] Schein, B.M., Corrigenda to “Commutative semigroups where congruences form a chain”,Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 23 (1975), 1247–1248.[21] Tamura, T.: Commutative semigroups whose lattice of congruences is a chain, Bull. Soc.Math. France 97 (1969), 369–380.Peter R. JonesDepartment of Mathematics, Statistics and Computer Science,Marquette University,Milwaukee, WI 53201, USAemail: peter.jones@marquette.eduAlexander L. PopovichDepartment of Mathematics and Computer Science,Ural Federal University,620000 Ekaterinburg, Russiaemail: alexanderpopovich@urfu.ru7

On Congruence Lattices of Nilsemigroups

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On Congruence Lattices of Nilsemigroups

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