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Idempotents and one-sided units: Lattice invariants and a semigroup of functors on the category of monoids

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Pith's one-line read This paper proves that the lattice of submonoids generated by idempotents and one-sided units in any monoid is determined by a four-bit type, and that the functors extracting these submonoids compose to form a monoid of exactly fifteen…

desk verdict A genuinely useful classification of idempotent/unit-generated submonoids with a 15-element functor monoid, mostly tight proofs, and only minor blemishes; worth refereeing. read the letter →

arxiv 1908.08225 v2 pith:3SP5BKHY submitted 2019-08-22 math.GR math.CTmath.RA

James East This is my paper · ORCID
classification math.GRmath.CTmath.RA MSC 20M5020M1020M1520M2018D10
keywords monoidsidempotentsunitsone-sidedlatticesfunctorsinvariantsGreen'srelations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Every monoid contains a small family of natural submonoids: subsets closed under the operation that are generated by idempotents (elements equal to their own square), by left or right units (elements with one-sided inverses), by two-sided units, and by combinations of these. This paper proves that the inclusion pattern among these submonoids, the lattice $L(M)$, is completely determined by four yes/no answers collected in a type $T(M)=(T_1(M),T_2(M),T_3(M),T_4(M))$, and that only a short finite list of lattice shapes can occur. It further shows that the operators choosing these submonoids are functors on the category of monoids and generate, under composition, a monoid of exactly fifteen functors. All sixteen possible types are realized by explicit monoids, so the classification is exhaustive rather than an artifact of missing examples. If correct, this means the collection of idempotent and unit submonoids is always one of a few rigid patterns fixed by four bits of data.

What carries the argument

The load-bearing mechanism is the four-bit type $T(M)$ together with the stable/unstable dichotomy from Green's relations. The identity element $1$ of a monoid is stable exactly when the $\mathcal{J}$-class of $1$ equals the unit group $G(M)$, equivalently when $M$ contains no bicyclic submonoid; in that case $G=G_L=G_R=G_{LR}$ and the lattice collapses to the five-element chain of Figure 3. When $1$ is unstable, Lemmas 3.5 and 3.6 force the seven submonoids $G,G_L,G_R,F,F_L,F_R,F_{LR}$ to remain distinct while leaving only the three binary choices measured by $T_2,T_3,T_4$. The argument's key identity is the imported product description $F_R(M)=G_R(M)\cdot E(M)$, which feeds into Lemma 3.5(ii) to show $G_L\cap F_R\subseteq G$; that containment is what prevents uncontrolled collapse and lets the type alone determine the whole lattice. The direct-product multiplicativity of the type, $T(M\times N)=T(M)\times T(N)$, then lets the paper realize all sixteen types from four small monoids.

What would settle it

Enumerate all monoids of order up to 8, compute for each monoid the type $T(M)$ and the lattice $L(M)$, and check that $L(M)$ is the diagram prescribed by Theorem 4.4; any monoid whose lattice is not the predicted shape is a direct counterexample. A more targeted test is to check whether every element of $F_R(M)$ has the form $g\,e_1\cdots e_k$ with $g$ a right unit and each $e_i$ idempotent, since the imported product description is the step where the argument would first fail.

Watch

Extended reading notes

Core claim

The paper's central claim is that the lattice $L(M)=\{X(M):X\in \mathcal{F}\}$, where $\mathcal{F}=\{O,E,G,G_L,G_R,G_{LR},F,F_L,F_R,F_{LR},I\}$, is classified by the binary quadruple $T(M)=(T_1(M),T_2(M),T_3(M),T_4(M))$. The four bits ask whether $G(M)=G_L(M)$, whether $F_{LR}(M)=M$, whether $F_{LR}(M)=G_{LR}(M)$, and whether $G(M)=\{1\}$; Theorem 4.4 states that for a stable identity the lattice is one of the eight diagrams in Figure 4, and for an unstable identity one of the eight diagrams in Figure 6, with Figure 7 listing the possibilities up to isomorphism. The same section shows every one of the sixteen quadruples arises by taking direct products of four basic monoids. Independently, Section 5 proves that the enlarged collection $\mathcal{F}^+=\mathcal{F}\cup\{Q,P,P_L,P_R\}$, with $Q=E\circ G_{LR}$, $P=F\circ G_{LR}$, $P_L=F_L\circ G_{LR}$, $P_R=F_R\circ G_{LR}$, is closed under composition and has exactly fifteen elements, the four new functors being genuinely new; the enhanced lattice $L^+(M)$ is then shown to be classified by the same four-bit type and hence to add no discriminating power beyond $L(M)$.

Load-bearing premise

Everything rests on the imported lemma that in every monoid the submonoid generated by right units and idempotents consists exactly of products of one right unit with finitely many idempotents; if some monoid failed that description, the proof that left units meet it only in the two-sided units would break, and with it the stable/unstable collapse analysis and the classification.

Editorial extensions

If this is right

  • For any monoid $M$, the lattice $L(M)$ has one of the finitely many shapes in Figure 7; deciding which one requires only the four bits of $T(M)$.
  • Every one of the sixteen types occurs, so the classification cannot be sharpened by adding further binary conditions of the same kind to the type.
  • The eleven functors of $\mathcal{F}$ together with $Q,P,P_L,P_R$ form a 15-element monoid under composition; no further new functors appear when all compositions are taken.
  • The enhanced lattice $L^+(M)$ is classified by the same four-bit type as $L(M)$, so it adds no new discriminating information beyond $L(M)$.
  • The monoid $\mathcal{F}^+$ is $\mathcal{J}$-trivial, so its Green's relations coincide with equality and the divisibility order in Figure 9 describes the whole structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same direct-product recipe that realizes all sixteen types also serves as a practical way to construct monoids with prescribed lattice shapes from a group, an idempotent-generated monoid, the positive integers, and the bicyclic monoid.
  • This suggests testing whether other natural element classes, for example regular elements or elements whose powers eventually repeat, give operators that again form a finite monoid under composition; the closure to fifteen here makes such finiteness plausible.
  • Remark 6.5 leaves open whether $L^+(M)$ is always a sublattice of $Sub(M)$; that can be settled by a computer search over small monoids, and a counterexample would show the enhanced lattice has a subtle incompleteness invisible to the four-bit classification.
  • The four added functors are best read as bookkeeping that closes the operator monoid under composition rather than as new invariants, since they change neither the type nor the discriminating power of the lattice.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. This paper studies the submonoids of an arbitrary monoid M generated by idempotents and by one- or two-sided units, viewed as functors on the category of monoids with composition as the operation; the basic functors are O, E, G, GL, GR, GLR, F, FL, FR, FLR, I. The main results are threefold. First, the lattice L(M) = {X(M) : X in F} is classified: a four-bit type T(M) records whether G = GL, whether FLR = M, whether FLR = GLR, and whether G = {1}; Proposition 4.2 shows that all sixteen types occur via direct products of a nontrivial group, an idempotent-generated monoid, the positive integers, and the bicyclic monoid, and Theorem 4.4 shows that the type completely determines L(M), whose possible shapes are the finite list in Figures 4, 6 and 7. Second, the monoid F+ generated by F together with the four new functors Q = E∘GLR, P = F∘GLR, PL = FL∘GLR, PR = FR∘GLR is proved to have exactly fifteen elements (Proposition 5.6), with a complete composition table (Table 3) and a description of its Green structure, including J-triviality and subsemigroup and congruence counts obtained with GAP (Section 5.5). Third, Section 6 classifies the enhanced lattice L+(M) = {X(M) : X in F+} and shows that it equals L(M) unless T1 = T3 = 0, in which case at most four additional nodes appear; the enhanced invariant carries no more isomorphism-type information than L.

Significance. If correct, the results give a complete and very small classification: every monoid's lattice of submonoids generated by idempotents and units is one of finitely many shapes determined by four yes/no questions, and the composition of any two of the fifteen functors is again one of them, so the associated operators form a 15-element J-trivial monoid. The paper's main strength is its explicitness: the composition tables are fully displayed; the distinctness claims are witnessed by four concrete monoids; all sixteen types are realized by explicit products; and the classification is falsifiable in that any monoid can be checked against its type. The paper is also honest about the limits of the invariant, including the open question of whether L+(M) is always a sublattice of Sub(M) (Remark 6.5). I explicitly checked the point most plausibly load-bearing, the identity FR(M) = GR(M)·E(M) imported from [22, Lemma 2.5] and used in Lemma 3.5(ii); it is elementary (conjugate idempotents past right units) and is not a gap. The GAP-based structural claims in Section 5.5 are stated clearly but would benefit from an accompanying script for full reproducibility.

minor comments (6)
  1. [Section 4, before (4.3)] In the unstable case, the text says that Lemma 3.7(ii) gives {G, GL, GR} ∩ {F, FL, FR, FLR} ≠ ∅; since it has just been established that E ≠ {1}, Lemma 3.7(ii) gives the opposite, namely that this intersection is empty, and the empty intersection is exactly what is needed to justify the distinctness of the seven submonoids listed in (4.3). This looks like a typographical error in the inequality symbol rather than a mathematical gap, but it should be corrected.
  2. [Lemma 3.5(ii)] The equality FR(M) = GR(M)·E(M) is imported from [22, Lemma 2.5] and is load-bearing, since it is used to prove that GL ∩ FR ⊆ G, which underlies the stable/unstable dichotomy (Lemma 3.10) and hence the classification in Theorem 4.4. The proof is very short (for a right unit g with right inverse g' and an idempotent e, one has eg = g(g'eg) with g'eg idempotent, so right units can be moved to the left of any product of idempotents); please include it so that the central argument is self-contained.
  3. [Section 5.5] The structural claims about F+ (J-triviality, 2904 subsemigroups, 1613 congruences, 76 principal congruences) are attributed to GAP, but no code or session output is provided, so these claims cannot be checked by a reader. Please include the GAP script or an explicit machine-readable verification artifact in an appendix or as supplementary material.
  4. [Proposition 5.6] The proof that |F+| = 15 rests on the assertion that L+(M) has size 15 for M = G×E×P×B, followed by the phrase that one may easily check this. Since this is the only place where the fifteen functors are shown pairwise distinct, please display the table of the fifteen quadruples (X(G), X(E), X(P), X(B)) or spell out which pairs of functors are separated by which factor; the preceding sentence on the lower bound |F+| ≥ 11 is also too compressed, as the point is that the eleven functors of F already give eleven distinct submonoids on a monoid of type (0,0,0,0).
  5. [Sections 2.1 and 3] The symbol E is used for both the set of idempotents and the submonoid they generate, and in Lemma 3.3 and in the proof of Lemma 3.5(ii) the two uses appear close together. Please adopt a clearer convention (for instance a distinct symbol for the set of idempotents) or state the convention once in Section 2.1, since this is a genuine source of possible misreading.
  6. [Theorem 4.4 and Figures 4, 6, 7] The classification proof is a prose case analysis in which the figures carry much of the information. Please state explicitly that the displayed Hasse diagrams are verified by combining Lemmas 3.5-3.10 with the type conditions, in particular that every displayed cover is a proper containment and that no containments other than the transitive closure of the displayed edges occur; this would make the role of the figures fully checkable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central lattice classification and 15-element functor monoid are established by explicit lemmas and constructions; citations to [22] are prior proved results, not restatements of the target claims.

full rationale

The derivation chain is not circular. The paper's central results -- that the lattice L(M) is determined by the four-bit type T(M), and that the functor monoid F+ has size 15 -- are proved by direct lattice-theoretic arguments, composition tables, and explicit separating examples. Where the paper imports results from the author's earlier article [22], notably [22, Lemma 2.1] in Lemma 3.3(i), [22, Lemma 2.3] in Lemma 3.10, and [22, Lemma 2.5] in Lemma 3.5(ii), these are prior published facts about arbitrary monoids, not restatements of the present classification. In particular, [22, Lemma 2.5] (FR = GRE) is used to prove GL ∩ FR ⊆ G; it is not derived from the target theorem and is independently verifiable by bubbling right units left past idempotents, so it is not an unverified premise smuggled in by self-citation. The type questions T1-T4 record genuine lattice and unit conditions, and Theorem 4.4 uses Lemmas 3.6, 3.7, and 3.10 to rule out further collapses; no fitted parameter is renamed as a prediction. Proposition 5.6 separates all 15 functors using the explicit monoid G × E × P × B and the composition table, again by construction. Two non-circular blemishes should be noted: before (4.3) the intersection {G, GL, GR} ∩ {F, FL, FR, FLR} is printed as nonempty when the argument and the subsequent distinctness claim require it to be empty, a typo; and the paper relies on [22] for a few elementary lemmas instead of reproving them in-line. Neither amounts to circularity. The open question in Remark 6.5 about whether L+(M) is always a sublattice of Sub(M) is a clearly stated limitation, not a hidden circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 4 invented entities

The paper introduces no free parameters and no data-fitting. Its dependencies are standard semigroup theory (Green's relations, stability, bicyclic monoid characterization) and a set of elementary lemmas from the author's prior article [22], which are cited as black boxes. The only new entities are four functors Q, P, PL and PR, but these are constructed explicitly as compositions of previously defined functors and are evaluated on concrete monoids in Table 4.

assumptions (5)
  • domain assumption The paper assumes, without proof here, several elementary results from its own prior article [22]: Lemmas 2.1, 2.3, 2.5, 2.8 and 2.9 about idempotents and one-sided units.
    These lemmas are cited as black boxes (e.g., in Lemmas 3.3, 3.5, 3.6, 3.10 and in Section 5.1) and are load-bearing for the collapse analysis that underlies the classification.
  • domain assumption The submonoid generated by idempotents and right units equals the set product GR * E ([22, Lemma 2.5]); used in Lemma 3.5(ii).
    This equality is used to prove GL ∩ FR is contained in G, a key step in the stable/unstable dichotomy and in the lattice classification.
  • standard math Green's relations and stability facts from [50], [64] and [25], including that the identity is stable iff J1 = H1; used in Lemma 3.10.
    Standard semigroup theory background that the paper invokes without proof to connect unit submonoids to Green's classes and stability.
  • standard math A monoid generated by x and y with yx = 1 and xy ≠ 1 is isomorphic to the bicyclic monoid ([45, pp. 31-32]); used in Lemma 3.6(ii) => (v).
    This classical characterization is used to show that G ≠ GL forces infinitely many idempotents in GLR, a structural fact in the collapse dichotomy.
  • standard math A finite join-semilattice with a bottom element is a lattice; used in Propositions 2.3 and 6.1.
    Standard lattice theory that lets the paper conclude L(M) and L+(M) are lattices once closure under joins is shown.
invented entities (4)
  • Q = E ∘ GLR independent evidence
    purpose: Fills the missing composition E ∘ GLR; produces the additional submonoid Q(M) = E(GLR(M)) in the enhanced lattice L+(M).
    Defined explicitly as a composition of two previously defined functors; its values on groups, idempotent-generated monoids, P, B and B0 are tabulated in Table 4, giving concrete checkable behavior.
  • P = F ∘ GLR independent evidence
    purpose: Fills the missing composition F ∘ GLR; produces the additional submonoid P(M) = F(GLR(M)).
    Defined explicitly as a composition of known functors; its values on the standard example monoids are listed in Table 4.
  • PL = FL ∘ GLR independent evidence
    purpose: Fills the missing composition FL ∘ GLR; produces the additional submonoid PL(M) = FL(GLR(M)).
    Defined explicitly as a composition of known functors; its values on the standard example monoids are listed in Table 4.
  • PR = FR ∘ GLR independent evidence
    purpose: Fills the missing composition FR ∘ GLR; produces the additional submonoid PR(M) = FR(GLR(M)).
    Defined explicitly as a composition of known functors; its values on the standard example monoids are listed in Table 4.

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Pith. "Pith review of Idempotents and one-sided units: Lattice invariants and a semigroup of functors on the category of monoids." pith.science (2026). https://pith.science/paper/3SP5BKHY

@misc{pith2026190808225,
  author       = {Pith},
  title        = {Pith review of: Idempotents and one-sided units: Lattice invariants and a semigroup of functors on the category of monoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SP5BKHY}},
  note         = {Machine review of arXiv:1908.08225}
}
abstract

For a monoid $M$, we denote by $\mathbb G(M)$ the group of units, $\mathbb E(M)$ the submonoid generated by the idempotents, and $\mathbb G_L(M)$ and $\mathbb G_R(M)$ the submonoids consisting of all left or right units. Writing $\mathcal M$ for the (monoidal) category of monoids, $\mathbb G$, $\mathbb E$, $\mathbb G_L$ and $\mathbb G_R$ are all (monoidal) functors $\mathcal M\to\mathcal M$. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We show that this monoid has size $15$, and describe its algebraic structure. We also show how to associate certain lattice invariants to a monoid, and classify the lattices that arise in this fashion. A number of examples are discussed throughout, some of which are essential for the proofs of the main theoretical results.

Figures

Figures reproduced from arXiv: 1908.08225 by the authors.

Figure 1
Figure 1. The generic shape of the lattice L (M). In general these submonoids need not be distinct. It will also be convenient to record the following obvious fact. For a monoid M, we write M0 for the monoid obtained by adjoining a new zero element 0 to M. Lemma 2.2. For any monoid M we have X(M0 ) = ( X(M) if X is one of O, G, GL, GR or GLR X(M) ∪ {0} if X is one of I, E, F, FL, FR or FLR. 2.3 Lattices For a monoid M, we wri… view at source ↗
Figure 2
Figure 2. The lattices L (B) and L (B0 ), where B is the bicyclic monoid. In both diagrams, the nodes represent distinct submonoids. which does not belong to L (B0 ). Of course, the submonoids FL(B0 ) and GLR(B0 ) do have a meet in L (B0 ) itself, as the latter is a lattice, but this meet in L (B0 ) is GL(B0 ) = hai; cf [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. The generic shape of the lattice L (M) when M has a stable identity. In general, some of the submonoids pictured in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: The lattice L (M) when M has a stable identity, according to the type T(M) = (1, i, j, k). In each case, the nodes represent distinct submonoids of M. These submonoids are shaded red in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: The generic shape of the lattice L (M) when M has an unstable identity. The submonoids shaded red are distinct, and thick lines indicate proper containment. ◦ O E G GL GR GLR F FL FR FLR I O O O O O O O O O O O O E O E O O O E E E E E G O O G G G G G G G G G GL O O G G…
Figure 6
Figure 6. Figure 6: The lattice L (M) when M has an unstable identity, according to the type T(M) = (0, i, j, k). In each case, the nodes represent distinct submonoids of M. Lemma 5.1. If N is a submonoid of M, and if GL(M), GR(M) ⊆ N, then X(N) = X(M) for X = G, GL, GR, GLR. Proof. We fi…
Figure 7
Figure 7. Figure 7: The possible lattices L (M) for a monoid M, up to lattice isomorphism. For X = FLR we have GLR ⊇ FLR(GLR) = E(GLR) ∨ GLR(GLR) = E(GLR) ∨ GLR = GLR, where we again used Lemma 5.1 in the third step. Thus, FLR(GLR) = GLR: i.e., FLR ◦GLR(M) = GLR(M). (ii). This is again cl…
Figure 8
Figure 8. Figure 8: The generic shape of the lattice L +(M). In general these submonoids need not be distinct. 5.3 More compositions Now that we have enlarged our list of functors to F+, we have a number of further compositions to calculate, namely those of the form X◦Y and Y◦X for X ∈ F+…
Figure 9
Figure 9. Figure 9: The divisibility order in the monoids F+ and E(F+): left and right, respectively. Proposition 6.1. For any monoid M, the set L +(M) is a finite ∨-subsemilattice of Sub(M), with top element I(M) = M and bottom element O(M) = {1}. Consequently, L +(M) is a lattice. Proof…
Figure 10
Figure 10. Figure 10: The generic shape of the lattice L +(M) when M has type T(M) = (0, i, 0, j). The submonoids shaded red are distinct, and thick lines indicate proper containment. The exact shape of L +(M) depends on the values of i = T2(M) and j = T4(M), and these determine which thin…
Figure 11
Figure 11. Figure 11: The lattice L +(M) when M has type T(M) = (0, i, 0, j). In each case, the nodes represent distinct submonoids of M. For other types we have L +(M) = L (M); cf. Figures 4 and 6. Remark 6.5. Recall that B0 is the bicyclic monoid with a zero adjoined. We noted in Remark …
Figure 12
Figure 12. Figure 12: The possible lattices L +(M) for a monoid M, up to lattice isomorphism. [24] J. East and R. D. Gray. Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals. J. Combin. Theory Ser. A, 146:63–128, 2017. [25] J. East and P. M. Higgins. Gree…

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