Arrow Relations in Lattices of Integer Partitions

Throughout the text we write $\mathbb{N}\mathrel{\mathop{:}}=\left\{0,1,2,\ldots\right\}$ for the set of natural numbers, and we set $\mathbb{N}_{+}\mathrel{\mathop{:}}=\mathbb{N}\setminus\left\{0\right\}$. An ordered set is a pair $\mathbb{P}=\left(P,\leq\right)$ where $P$ is a set and ${\leq}\subseteq P\times P$ is a reflexive, antisymmetric and transitive binary relation on it. For $a,b\in P$ we have $a\geq b$ if and only if $b\leq a$, and we write $a<b$ for $a\leq b$ and $a\neq b$ as usual. Moreover, we say that $a$ is covered by $b$ in $\mathbb{P}$ (or $b$ covers $a$ in $\mathbb{P}$), symbolically $a\prec b$, if $a<b$ and $a\leq x<b$ implies $x=a$ for each $x\in P$. The dual of $\mathbb{P}$ is the ordered set $\left(P,\geq\right)$. An order-isomorphism $\varphi\colon\mathbb{P}\to\mathbb{Q}$ is given by a map $\varphi\colon P\to Q$ such that for all $a,b\in P$ the inequality $a\leq b$ in $\mathbb{P}$ is equivalent to $\varphi(a)\leq\varphi(b)$ in $\mathbb{Q}$. The ordered sets $\mathbb{P}$ and $\mathbb{Q}$ are then said to be (order-)isomorphic. An order-antiisomorphism or dual isomorphism $\varphi\colon\mathbb{P}\to\mathbb{Q}$ is an order isomorphism between $\mathbb{P}$ and the dual of $\mathbb{Q}$, and $\mathbb{P}$ and $\mathbb{Q}$ are then called antiisomorphic or dually isomorphic; $\mathbb{P}$ is self-dual (or autodual) if it is dually isomorphic to itself. Furthermore, we define $\mathop{\downarrow}x\mathrel{\mathop{:}}=\left\{y\in\mathbb{P}\mid y\leq x\right\}$ as the principal down-set of $x\in P$, and $\mathop{\uparrow}x\mathrel{\mathop{:}}=\left\{y\in\mathbb{P}\mid x\leq y\right\}$ as the principal up-set of $x$.

A complete lattice is an ordered set $\mathbb{L}=(L,\leq)$ where any subset $S\subseteq L$ has a greatest common lower bound $\bigwedge S\in L$ (called infimum of $S$) and a least common upper bound $\bigvee S\in L$ (called supremum of $S$). It is customary to write $a_{1}\wedge\dotsm\wedge a_{k}$ for $\bigwedge\left\{a_{1},\dotsc,a_{k}\right\}$ and $a_{1}\vee\dotsm\vee a_{k}$ for $\bigvee\left\{a_{1},\dotsc,a_{k}\right\}$ for finitely many elements $a_{1},\dotsc,a_{k}\in L$. A subset $D\subseteq L$ is called (completely) join-dense in $\mathbb{L}$ if for every $x\in L$ there is a subset $S\subseteq D$ with $x=\bigvee S$; $D$ is (completely) meet-dense in $\mathbb{L}$ if for each $x\in L$ there is some $S\subseteq D$ with $x=\bigwedge S$. We say that $a\in L$ is completely join-irreducible, denoted by $\bigvee$-irreducible, if for all $S\subseteq L$ such that $a=\bigvee S$ we necessarily have that $a\in S$. Dually, $a\in L$ is completely meet-irreducible, or $\bigwedge$-irreducible, if for all $S\subseteq L$ such that $a=\bigwedge S$ we must have $a\in S$. For a finite non-empty lattice, it is sufficient to check this condition for the empty and all two-element subsets $S$ of $L$. That is, for finite $L\neq\emptyset$, an element $a\in L$ is completely join-irreducible, if $a$ is not the minimum element of $L$ and for all $b,c\in L$ the condition $a=b\vee c$ implies that $a=b$ or $a=c$. This happens exactly if $a$ covers exactly one element below it, cf. [10, Proposition 2]. Dually, for a finite lattice, $a\in L$ is completely meet-irreducible if it is not the top element of $L$ and for all $b,c\in L$ with $a=b\wedge c$ it follows that $b=a$ or $c=a$. This condition is met precisely if $a$ has exactly one upper cover. For a complete lattice $\mathbb{L}$, we denote by $\mathcal{J}\left(\mathbb{L}\right)$ and by $\mathcal{M}\left(\mathbb{L}\right)$ the sets of all completely join-irreducible elements, and of all completely meet-irreducible elements of $\mathbb{L}$, respectively. Elements of $\mathcal{J}\left(\mathbb{L}\right)\cap\mathcal{M}\left(\mathbb{L}\right)$ are called doubly completely irreducible. In the lattice diagrams shown in Fig. 3 and 6, the completely irreducible elements have been highlighted using half or completely filled nodes.

A (complete) lattice $\mathbb{L}=(L,{\leq})$ is supremum-founded if, for any two $x<y$ from $L$, the set $\left\{\left.p\in L\ \vphantom{p\leq y\mathbin{\&}p\nleq x}\right|\ p\leq y\mathbin{\&}p\nleq x\right\}$ contains a $\leq$-minimal element; the dual property that for any $x<y$ in $L$ the set $\left\{\left.p\in L\ \vphantom{x\leq p\mathbin{\&}y\nleq p}\right|\ x\leq p\mathbin{\&}y\nleq p\right\}$ includes a $\leq$-maximal element is called infimum-founded. The lattice $\mathbb{L}$ is doubly founded if it is both supremum-founded and infimum-founded, see [10, p. 33]. Every chain-finite and hence every finite lattice is doubly founded, cf. [10, p. 33 et seqq., Fig. 1.11, p. 35].

Formal concept analysis (FCA) is a theoretical framework that harnesses the powers of general abstract Galois theory and the structure theory of complete lattices for data analysis and many other applications. At its core lies the notion of a Galois connection between (the power sets of) sets $G$ and $M$, induced by a binary relation $I\subseteq G\times M$. This data is collected in a formal context $\mathbb{K}=(G,M,I)$, and the elements of $G$ and $M$ are given the interpretative names objects and attributes, respectively. The set $I$ is called the incidence relation, and $(g,m)\in I$ is usually written as $g\mathrel{I}m$ and read as ‘object $g$ has attribute $m$’. When $G$ and $M$ are finite, the context $\mathbb{K}$ is often given as a cross table, where the objects form the rows, the attributes label the columns, and the crosses represent the characteristic function of $I$ on $G\times M$. Formal concept analysis extends the ‘prime notation’ for the Galois derivatives, which is common in classical Galois theory [12, Chapter V, Theorem 2.3 et seqq.], to general formal contexts $K=(G,M,I)$. For every set $A\subseteq G$ of objects, $A^{\prime}\mathrel{\mathop{:}}=\left\{m\in M\mid\forall g\in A\colon g\mathrel{I}m\right\}$ assigns to it the set of attributes commonly shared by all objects of $A$. Dually, for $B\subseteq M$, the set $B^{\prime}\mathrel{\mathop{:}}=\left\{g\in G\mid\forall m\in B\colon g\mathrel{I}m\right\}$ contains exactly those objects possessing all the attributes in $B$. A formal concept is a pair $(A,B)$ where the extent $A\subseteq G$ and the intent $B\subseteq M$ are sets that are mutually Galois closed: $A=B^{\prime}$ and $B=A^{\prime}$. Intents of the form $B=\left\{g\right\}^{\prime}$ with $g\in G$ are called object intents and are written as $g^{\prime}$, for short; dually extents $A=\left\{m\right\}^{\prime}\mathrel{\mathopen{=}{\mathclose{:}}}m^{\prime}$ with $m\in M$ are referred to as attribute extents. The equivalent conditions $A_{1}\subseteq A_{2}$ and $B_{1}\supseteq B_{2}$ define an order $(A_{1},B_{1})\leq(A_{2},B_{2})$ on the set $\mathfrak{B}(\mathbb{K})$ of all formal concepts. $\underline{\mathfrak{B}}(\mathbb{K})=(\mathfrak{B}(\mathbb{K}),\leq)$ becomes a complete lattice under this order, the concept lattice of $\mathbb{K}$. The fundamental theorem of formal concept analysis [10, Theorem 3] states that, in fact, every complete lattice is a concept lattice, up to isomorphism. Namely, if $\mathbb{L}$ is a complete lattice, then $\mathbb{L}\cong\underline{\mathfrak{B}}(L,L,{\leq})$. For finite lattices, which are always complete, this construction can be improved: $\mathbb{L}\cong\underline{\mathfrak{B}}(\mathbb{K})$, where $\mathbb{K}=(\mathcal{J}\left(\mathbb{L}\right),\mathcal{M}\left(\mathbb{L}\right),{\leq})$ is the standard context of $\mathbb{L}$ [10, Proposition 12]. This applies to partition lattices $\mathcal{L}_{n}$ in particular.

A central notion for this paper will be the arrow relations of a formal context, which fill up some of the empty cells in the cross table.

We will now derive a useful characterisation of the arrow relations in standard contexts of doubly founded lattices.

The following sufficient condition will lead to our main tool to establish arrow relations in standard contexts, in particular in $\mathbb{K}(\mathcal{L}_{n})$.

We shall need the following statement for infimum/supremum founded complete lattices, which can be read from [10, Fig. 1.11, p. 35]. For completeness, we provide a proof of this fundamental fact.

For our purposes the following characterisation of the arrow relations is the most appropriate one since the lattice $\mathcal{L}_{n}$ is finite, thus all its chains have only a finite number of elements, and it hence is doubly founded.

In Section 1 we explained that so-called arrow-closed subcontexts (of the standard context) are a key ingredient in order to obtain subdirect decompositions of finite lattices. We now provide concrete definitions as far as they are needed in this paper. A subcontext of a context $\mathbb{K}=(G,M,I)$ is a context $\tilde{\mathbb{K}}=(H,N,J)$ where $H\subseteq G$, $N\subseteq M$ and $J=I\cap(H\times N)$. For a clarified context $\mathbb{K}$, that is, $g^{\prime}=h^{\prime}$ implies $g=h$, and $m^{\prime}=n^{\prime}$ implies $m=n$ for all $g,h\in G$ and $m,n\in M$, such a subcontext $\tilde{\mathbb{K}}$ is arrow-closed if for all $h\in H$, $m\in M$, $n\in N$ and $g\in G$ the condition $g\mathrel{\nearrow}m$ implies $m\in N$, and $g\mathrel{\swarrow}n$ implies $g\in H$, see [10, Definition 46]. Note that the standard context of a finite lattice $\mathbb{L}$ is always clarified and reduced [10, Proposition 12]. For a finite clarified context $\mathbb{K}=(G,M,I)$ and $G_{1}\subseteq G$ and $M_{1}\subseteq M$ there is always a smallest arrow-closed subcontext $\tilde{\mathbb{K}}=(H,N,I\cap(H\times N))$ of $\mathbb{K}$ with $G_{1}\subseteq H$ and $M_{1}\subseteq N$. It can be obtained by constructing the directed graph $\left(G\mathbin{\dot{\cup}}M,{\mathrel{\nearrow}}\cup{(\mathrel{\swarrow})^{-1}}\right)$ and considering the (not necessarily strongly) connected directed components $[x]$ of each $x\in G_{1}\mathbin{\dot{\cup}}M_{1}$. One then forms $\bigcup_{x\in G_{1}\mathbin{\dot{\cup}}M_{1}}[x]$, which can be written in a unique way as $H\mathbin{\dot{\cup}}N$ with $H\subseteq G$ and $N\subseteq M$. In particular, starting from $G_{1}=\left\{g\right\}$ and $M_{1}=\emptyset$ with $g\in G$ (or dually from $G_{1}=\emptyset$ and $M_{1}=\left\{m\right\}$ with $m\in M$), we get the one-generated arrow-closed subcontexts of $\mathbb{K}$, cf. [10, Section 4.1]. Note that if the context $\mathbb{K}$ is reduced, we may always concentrate on using either only objects or attributes for constructing all its one-generated arrow-closed subcontexts.

Our aim is to study the arrow relations of the standard context of the lattice $\mathcal{L}_{n}$ of positive integer partitions, which is formed by the sets $\mathcal{J}\left(\mathcal{L}_{n}\right)$ and $\mathcal{M}\left(\mathcal{L}_{n}\right)$. First, we define formally, what a partition and the dominance order is.

For example, $(5,4,1,1,0,0,0,0,0,0,0)$ is a partition of $11$ because ${5\geq 4\geq 1\geq 0}$ and $5+4+1+1=11$. By removing the trailing zeros we can represent it more compactly as $(5,4,1,1)$. Graphically, we can illustrate a partition using a diagram drawn with small squares or ‘bricks’ arranged in a downward ladder shape (cf. Fig. 2), which is known as Ferrers diagram (usually Ferrers diagrams are drawn rotated clockwise by a $90$ degrees angle, but the chosen presentation is more useful to us, cf. Definition 2.9). Given a partition $a$ of $n$, one obtains the conjugated or dual partition $a^{*}$ in the sense of [5] and [9] as $a^{*}=(a_{1}^{*},\dotsc,a_{n}^{*})$ where $a_{i}^{*}\mathrel{\mathop{:}}=\left\lvert\{1\leq j\leq n\mid a_{j}\geq i\}\right\rvert$ for all $i\in\left\{1,\dotsc,n\right\}$. The Ferrers diagram of $a^{*}$ can be seen from the diagram of $a$ by reading it by rows, from bottom to top. For instance, the partition $(5,4,1,1)$ of $11=5+4+1+1$ has the Ferrers diagram shown in Fig. 2, and its conjugate consists of $4$ bricks from the first row, $2$ bricks from the second to fourth, and $1$ brick from the fifth row. We thereby obtain the partition $(4,2,2,2,1)$; its Ferrers diagram is also depicted in Fig. 2.

We denote the set of all partitions of an integer $n\in\mathbb{N}$ by $\operatorname{Part}(n)$. From the construction of the conjugate via the Ferrers diagram, it is easy to see that $\left(a^{*}\right)^{*}=a$ holds for every $a\in\operatorname{Part}(n)$. Therefore, partition conjugation is an involutive permutation of $\operatorname{Part}(n)$. One may order the set $\operatorname{Part}(n)$ in different ways, for example, lexicographically, or pointwise. In this paper we are interested in the dominance order established by Brylawski [5].

Brylawski showed in [5, Proposition 2.2] that the set $\operatorname{Part}(n)$ forms a lattice under the dominance order; further arguments for this were given in [6, Chapter 3] and [9, Section 2]. We denote by $\mathcal{L}_{n}$ the lattice of all partitions of $n\in\mathbb{N}$ with the dominance order. It follows from [5, Proposition 2.8] that for partitions $a,b\in\mathcal{L}_{n}$ we have $a\leq b$ if and only if $a^{*}\geq b^{*}$, that is, partition conjugation is an order-antiautomorphism of $\mathcal{L}_{n}$, making $\mathcal{L}_{n}$ is self-dual.

Brylawski in [5, Proposition 2.3] characterised precisely two possibilities for downward movement along the covering relation of $\mathcal{L}_{n}$. These were later given a more intuitive geometric interpretation as transition rules by Latapy and Phan [13, Fig. 2, p. 1358], which we are going to follow in this article. The subsequent definitions are required for this.

For example, the left Ferrers diagram in Fig. 2 has a cliff in the second position ($j=2$) and its dual has a cliff at $j=1$ and a slippery step (with $\ell=2$) at $j=4$.

Since the completely join-irreducible elements of $\mathcal{L}_{n}$ have exactly one lower cover, according to Lemma 2.10, the partitions in $\mathcal{J}\left(\mathcal{L}_{n}\right)$ are precisely those to which exactly one of the transition rules (1) or (2) applies. Therefore, the $\bigvee$-irreducible partitions of $n$ can be characterised as those that have exactly one cliff (and no slippery step) or exactly one slippery step (and no cliff). Based on this Brylawski [5] split the set $\mathcal{J}\left(\mathcal{L}_{n}\right)$ into four categories, extending this by conjugation to $\mathcal{M}\left(\mathcal{L}_{n}\right)$. To be able to express our results more compactly, in the following statement we have slightly modified the borders between the different types of irreducibles compared to [5], making them in particular disjoint.

Observe that for each $(L,J)\in\left\{(\ref{typeA},\ref{typeI}),(\ref{typeB},\ref{typeII}),(\ref{typeC},\ref{typeIII}),(\ref{typeD},\ref{typeIV})\right\}$ we have that if $g\in\mathcal{J}\left(\mathcal{L}_{n}\right)$ has type $L$, then $g^{*}$ has type $J$, and if $m\in\mathcal{M}\left(\mathcal{L}_{n}\right)$ has type $J$, then $m^{*}$ has type $L$. Therefore, the pairs $(L,J)$ of categories of completely irreducible elements of $\mathcal{L}_{n}$ are completely dual to each other.

We mentioned that $\mathcal{L}_{n}$ is a self-dual lattice under partition conjugation as involutive antiautomorphism, see [5, Proposition 2.8]. Using this fact we obtain the following simple but useful results to switch down and up-arrows.

Similarly, one can prove the following lemma.

Combining Lemma 2.12 and Lemma 2.13, we obtain the following corollary.

To obtain some familiarity with the completely irreducible elements of $\mathcal{L}_{n}$ and the characterisations concerning the arrow relations given in Subsection 2.2, we consider the lattice $\mathcal{L}_{7}$ shown in Fig. 3.