The congruence lattice of NI(X) for the exceptional finite sets $|X|=3$ and $|X|=5$

For a set $X$, $NI(X)$ denotes the subsemigroup of the symmetric inverse semigroup $I(X)$ generated by all nilpotent (some power equal to the empty map) partial injections. Marques-Smith and Sullivan (2006) described all congruences on $NI(X)$: for finite $X$ their Corollary 1 proves the congruence lattice is a chain, but only under the hypothesis $|X|\ge 6$, and the single small case $|X|=4$ is settled by a worked example. The remaining cases are the small odd finite cardinalities: since for even $n$ the source gives $NI(X)=I(X)$ with classically known congruences, the unworked cases are exactly $|X|=3$ and $|X|=5$ (plus the degenerate $|X|\le2$), where the source's normal-subgroup parametrisation of congruences on the top Rees factor $E_{n-1}\cup\{0\}$ breaks down because its proof invokes the simplicity of $\operatorname{Alt}(Y)\cong A_{n-1}$ on the $(n-1)$-subset $Y$, which requires $|Y|\ge5$: for $|X|=3$ the group is $A_2$, the trivial group, and for $|X|=5$ it is $A_4$, which is not simple. The problem asks to complete the classification: determine the congruence lattice of $NI(X)$ for $|X|=3$ and $|X|=5$, semigroups of orders 19 and 1126, and whether these lattices are chains, together with the congruences on the Rees factors $NI_{r+1}/NI_r$.

Background

Let $X$ be a set. The symmetric inverse semigroup $I(X)$ consists of all partial injective transformations of $X$ (one-to-one maps between subsets of $X$) under composition, with the empty map acting as a zero element. An element $a\in I(X)$ is nilpotent if $a^r$ is the empty map for some positive $r$; writing $r(a)$ for the rank of $a$ (the cardinality of its range), $NI(X)$ denotes the subsemigroup of $I(X)$ generated by all nilpotent elements. Sullivan (1987) proved that for infinite $X$ of cardinality $k$, an element $a\in I(X)$ is a product of nilpotents exactly when its defect $d(a)=|X\setminus\operatorname{ran}a|$ and gap $g(a)=|X\setminus\operatorname{dom}a|$ both equal $k$, so $NI(X)$ is an inverse semigroup (a regular subsemigroup whose idempotents commute). Gomes and Howie (1987) proved the finite analogue: for $|X|=n\ge3$, $NI(X)$ is an inverse semigroup, equal to all of $I(X)$ when $n$ is even, and to $I_{n-1}\cup E_{n-1}$ when $n$ is odd, where $I_{n-1}=\{a:r(a)<n-1\}$ and $E_{n-1}$ is the set of rank $n-1$ elements whose unique completion to a total permutation of $X$ is an even permutation; thus $E_{n-1}$ is closely tied to the alternating group $A_n$ on $X$.

A congruence on a semigroup is an equivalence relation compatible with multiplication; the set of all congruences forms a lattice under inclusion, and it is called a chain when any two congruences are comparable. The congruences of $I(X)$ itself are classical (Scheiblich 1973): they are built from the Rees congruences of the rank ideals together with normal subgroups of the symmetric groups attached at each rank level. Here, for an ideal $NI_r=\{a\in NI(X):r(a)<r\}$, the Rees congruence collapses $NI_r$ to a single point, and the Rees factor semigroup $NI_{r+1}/NI_r$ adjoins a zero to the set of elements of rank exactly $r$. Marques-Smith and Sullivan (1999) showed the $NI_r$ are precisely the proper ideals of $NI(X)$.

Building on that ideal structure, Marques-Smith and Sullivan (2006) described all congruences on $NI(X)$. Every non-identity, non-universal congruence $\rho$ has a primary rank $\eta(\rho)$, meaning that the $\rho$-class of the zero element is exactly $NI_r$ with $r=\eta(\rho)$. For finite primary rank, Theorem 5 of that paper (valid for every set $X$ and every positive integer $r$) reduces the problem to the Rees factors: $\rho=\sigma^+$ for a non-universal congruence $\sigma$ on $NI_{r+1}/NI_r$, where $\sigma^+$ is an explicit lifting construction from Lemma 6, again valid for arbitrary $X$. Lemma 7 then parametrises the non-universal congruences on $NI_{r+1}/NI_r$ by normal subgroups $N$ of the symmetric group $G(Y)\cong S_r$ on an $r$-subset $Y$ of $X$, but only under the hypothesis $|X|\ge6$: the proof passes through the simplicity of the alternating group $\operatorname{Alt}(Y)$, which requires $|Y|\ge5$, and thereby also handles the exceptional odd factor $E_{n-1}\cup\{0\}$. Consequently Corollary 1 — for finite $X$ with at least six elements, the congruence lattice of $NI(X)$ is a chain — is proved only for $|X|\ge6$; the paper treats the one small case $|X|=4$ by an explicit example exhibiting the full chain. For infinite $X$, Theorem 8 gives a necessary-and-sufficient description of all non-universal congruences by interlaced sequences of cardinals in the style of the classical description for $I(X)$, so the infinite case is complete as a classification.

Problem Statement

Complete the Marques-Smith–Sullivan classification of congruences on $NI(X)$ for the remaining small finite cardinalities. Since for even $n$ the source's Theorem 2(a) gives $NI(X)=I(X)$ with classically known congruences, and $n=4$ is settled by the source's worked example, the untreated cases are exactly the odd cardinalities $|X|=3$ and $|X|=5$, together with the degenerate $|X|\le2$ which fall outside the source's standing assumption $|X|\ge3$. Specifically: (i) determine all non-universal congruences on the Rees factor semigroups $NI_{r+1}/NI_r$ for $|X|=3$ and $|X|=5$ — in particular on the top factors $NI_3/NI_2=E_2\cup\{0\}$ and $NI_5/NI_4=E_4\cup\{0\}$, where the source's Lemma 7 argument cannot be invoked because its proof requires the simplicity of $\operatorname{Alt}(Y)\cong A_{n-1}$ on the $(n-1)$-subset $Y$, hence $|Y|\ge5$: for $|X|=3$ the group is $A_2$, the trivial group, and for $|X|=5$ it is $A_4$, which is not simple; and (ii) determine the complete congruence lattice of $NI(X)$, a semigroup of order 19 for $|X|=3$ and of order 1126 for $|X|=5$, and state whether each of these two lattices is a chain (the cases $|X|=4$ and $|X|\ge6$ are already known).

References

1. M. Paula O. Marques-Smith and R. P. Sullivan, 'Congruences on semigroups generated by injective nilpotent transformations', Bulletin of the Australian Mathematical Society 74 (2006), 393–409. DOI: 10.1017/S0004972700040454. https://doi.org/10.1017/S0004972700040454
2. M. P. O. Marques-Smith and R. P. Sullivan, 'Congruences on nilpotent-generated partial transformation semigroups', Algebra Colloquium 16 (2009), 229–242. DOI: 10.1142/S1005386709000236. https://doi.org/10.1142/S1005386709000236
3. M. Paula O. Marques-Smith and R. P. Sullivan, 'The ideal structure of nilpotent-generated transformation semigroups', Bulletin of the Australian Mathematical Society 60 (1999), 303–318. DOI: 10.1017/S0004972700036418. https://doi.org/10.1017/S0004972700036418
4. H. E. Scheiblich, 'Concerning congruences on symmetric inverse semigroups', Czechoslovak Mathematical Journal 23 (1973), 1–10. DOI: 10.21136/CMJ.1973.101138. https://doi.org/10.21136/CMJ.1973.101138
5. Gracinda M. S. Gomes and John M. Howie, 'Nilpotents in finite symmetric inverse semigroups', Proceedings of the Edinburgh Mathematical Society 30 (1987), 383–395. DOI: 10.1017/S0013091500026778. https://doi.org/10.1017/S0013091500026778
6. R. P. Sullivan, 'Semigroups generated by nilpotent transformations', Journal of Algebra 110 (1987), 324–343. DOI: 10.1016/0021-8693(87)90049-4. https://doi.org/10.1016/0021-8693(87)90049-4