Graded posets inverse zeta matrix formula

Andrzej Krzysztof Kwaśniewski

Member of the Institute of Combinatorics and its Applications

High School of Mathematics and Applied Informatics

Kamienna 17, PL-15-021 Białystok, Poland

e-mail: kwandr@gmail.com

Abstract:
We arrive at the explicit formula for the inverse of zeta matrix for any graded posets with the finite set of minimal elements following the first reference which is referred to as SNACK that is Sylvester Night Article on Cobweb posets and KoDAG graded digraphs. In SNACK the way to arrive at formula of the zeta matrix for any graded posets with the finite set of minimal elements was delivered and explicit form was given. We present here effective way toward the formula for the inverse of zeta matrix which is being unearthed via adjacency and zeta matrix description of bipartite digraphs chains, the representatives of graded posets with sine qua non essential use of digraphs and matrices natural join introduced by the present author.

Namely, the bipartite digraphs elements of such chains amalgamate so as to form corresponding cover relation graded poset digraphs with corresponding adjacency matrices being amalgamated throughout natural join constituting adequate special database operation.

As a consequence apart from zeta function also the Möbius function explicit expression for any graded posets with the finite set of minimal elements is being arrived at.

Purposely, on the way - special number theoretic code-triangles for KoDAGs are proposed and apart from the author combinatorial interpretation of -nomial coefficients another related interpretation is inferred while referring to the number of all maximal chains in the corresponding poset interval. The formula for August Ferdinand Möbius matrix is also interpreted combinatorially.

Key Words: graded digraphs, cobweb posets, natural join

AMS Classification Numbers: 06A06 ,05B20, 05C75

This is The Internet Gian-Carlo Rota Polish Seminar article

No 4, Subject 2, 2009-03-14 130-th Birthday of Albert Einstein

http://ii.uwb.edu.pl/akk/sem/sem_rota.htm

arXiv:0903.2575v1 [v1] Sat, 14 Mar 2009 20:32:01 130-th Birthday of Albert Einstein

## 1 Reference information and matrix formula.

1.1. The Upside Down Notation Principle

We shall here take for granted the notation and the results of [1] which is referred to as SNACK that is Sylvester Night Article on Cobweb posets and Kodag graded digraphs.

In particular denotes cobweb partial order set (cobweb poset) while denotes its incidence algebra over the ring . Correspondingly denotes arbitrary graded poset while denotes its incidence algebra over the ring .

For example might be taken to be Boolean algebra , the field , the ring of integers or real or complex or -adic fields. The present article is the next one in a series of papers listed in reversed order of appearence and these are: [1],[2],[3]. The authors upside down notation is used throughout this paper i.e. . The Upside Down Notation Principle used since last century effectively (see [1-20] and for earlier references therein; in particular see Appendix in [9] copied from [32]) may be formulated as a Principle i.e. trivial, powerful statement as follows. Through all the paper denotes a natural numbers valued sequence sometimes specified to be Fibonacci or others - if needed. Among many consequences of this is that graded posets ( their cover relation digraphs Hasse diagrams) are connected and sets of their minimal elements are finite.

Comment 0. Mantra

If the statement depends (relies, is based on,”’lies in ambush”’…..) only on the fact that is a natural valued numbers sequence
then if the statement is proved true for then it is true for any natural valued numbers sequence .

The Upside Down Notation Principle

1. Let the statement depends only on the fact that is a natural numbers valued sequence.

2. Then if one proves that is true - the statement is also true.

Formally - use equivalence relation classes induced by co-images of and proceed in a standard way.

1.2. Ponderables.

###### Definition 1

Let . Let . Let be the graded partial ordered set (poset) i.e. and constitutes ordered partition of . A graded poset with finite set of minimal elements is called cobweb poset iff

Note. By definition of being graded its levels are independence sets and of course partial order up there in Definition 1 might be replaced by .

The Definition 1 is the reason for calling Hasse digraph of the poset a KoDAG as in Professor Kazimierz Kuratowski native language one word Komplet means complete ensemble- see more in [3] and for the history of this name see: The Internet Gian-Carlo Polish Seminar Subject 1. oDAGs and KoDAGs in Company (Dec. 2008).

Simultaneously - for the history of the Kwaśniewski The Upside Down Notation Principle see: The Internet Gian-Carlo Polish Seminar Subject 2, upside down notation ; leitmotiv: Is the upside down notation efficiency - an indication? of a structure to be named? (Feb. 2009).

###### Definition 2

Let be an arbitrary natural numbers valued sequence, where . We say that the graded poset is denominated (encoded=labelled) by iff for . We shall also use the expression - ”‘-graded poset”’.

1.3. Combinatorial interpretation.

For combinatorial interpretation of cobweb posets via their cover relation digraphs (Hasse diagrams) called KoDAGs see [4,5]. The recent equivalent formulation of this combinatorial interpretation is to be found in [4] (Feb 2009) or [6] from which we quote it here down.

###### Definition 3

-nomial coefficients are defined as follows

while and with staying for falling factorial. is called -graded poset admissible sequence iff ( In particular we shall use the expression - -cobweb admissible sequence).

###### Definition 4

i.e. is the set of all maximal chains of

and consequently (see Section 2 in [9] on Cobweb posets’ coding via lattice boxes)

###### Definition 5

() Let

Note. The is the hyper-box points’ set [9] of Hasse sub-diagram corresponding maximal chains and it defines biunivoquely the layer as the set of maximal chains’ nodes (and vice versa) - for these arbitrary -denominated graded DAGs (KoDAGs included).

The equivalent to that of [4,5] formulation of combinatorial interpretation of cobweb posets via their cover relation digraphs (Hasse diagrams) is the following.

Theorem 1 [6,4]

(Kwaśniewski) For -cobweb admissible sequences -nomial coefficient is the cardinality of the family of equipotent to mutually disjoint maximal chains sets, all together partitioning the set of maximal chains of the layer , where .

For environment needed and then simple combinatorial proof see [4,5] easily accessible via Arxiv.

Comment 1. For the above Kwaśniewski combinatorial interpretation of -nomials’ array it does not matter of course whether the diagram is being directed or not, as this combinatorial interpretation is equally valid for partitions of the family of in comparability graph of the Hasse digraph with self-explanatory notation used on the way. The other insight into this irrelevance for combinatoric interpretation is [9]: colligate the coding of by hyper-boxes. (More on that soon). And to this end recall what really also matters here : a poset is graded if and only if every connected component of its comparability graph is graded. We are concerned here with connected graded graphs and digraphs.

For the relevant recent developments see [7] while [8] is their all source paper as well as those reporting on the broader research (see [9-20,22-26] and references therein). The inspiration for ”‘philosophy”’ of notation in mathematics as that in Knuth’s from [21] - in the case of ”‘upside-downs”’ has been driven by Gauss ”’-Natural numbers”’ from finite geometries of linear subspaces lattices over Galois fields. As for the earlier use and origins of the use of this author’s upside down notation see [27-43].

Comment 2. Colligate any binary relation with Hasse digraph cover relation and identify as in SNAC with incidence algebra zeta function and with zeta matrix of the poset associated to its Hasse digraph, where

The reflexive reachability relation is defined as

= transitive and reflexive closure of

where is the Boolean adjacency matrix of the relation simple digraph and stays for Boolean product.

Then colligate and/or recall from SNACK the resulting schemes. Schemes:

Remark 1. Obvious. Needed also for the next Section. Compare with the Observation 3. below.

The matrix ( the algebra structure coding element of the incidence algebra is the characteristic function of a partial order relation for any given - graded poset including - cobweb posets :

The consequent (customary-like notation included) notation of other algebra important elements then - for the any fixed order - is the following [3,2,1]:

Recall from SNACK : is the biadjacency i.e cover relation matrix of the adjacency matrix .

Note: biadjacency and cover relation matrix for bipartite digraphs coincide. By extension - we call cover relation matrix the biadjacency matrix too in order to keep reminiscent convocations going on.

As a consequence - quoting SNACK - we have:

or equivalently

In view of the all above the following is obvious;

except for the trivial case.

Anticipating considerations of Section III and customarily allowing for the identifications: - consider :

in order to note that ( )

= the number of all maximal chains in the poset interval

where and for , say , with the reflexivity (loop) convention adopted i.e. .

Sub-Remark 1.1. It is now a good - prepared for - place to note further relevant properties of constructs as to be used in the sequel. These are the following.

for while .

Let . Then for -cobweb posets (what about just -graded?) we note that

hence

for while

for while . Let us now see in more detail how this kind (Q.M.?) of mimics of Markov property is intrinsic for natural joins of digraphs. For that to do consider levels i.e. independent (stable) sets and extend the notation accordingly so as to encompass

Let

Then

for , … (for ?). In the case of cobweb posets (what about just -graded?) the numbers are the same for each therefore we have for cobwebs

which in view of is of course consistent with We consequently notice that - with self-evident extension of notation:

The frequently used block matrices are: 1) which denotes matrix of ones i.e. ; and , 2) and which stays for matrix of ones and zeros accordingly to the -graded poset has been fixed - see Observation 2.

In the block matrices language the above Markov property for cobweb posets (what about just -graded?) reads as follows (to be used in Section 2) for example :

Well, what about then just -graded? - See Comment 3 and its Warning.

Comment 3. Colligate and make identifications of graded DAGs with -ary relations as in SNAC:

for the natural join of di-bicliques and similarly for being natural join of any sequence binary relations

Warning. Note that not for all -graded posets their partial orders may be consequently identified with -ary relations, where while . This is possible iff no biadjacency matrices entering the natural join for has a zero column or a zero row. If a vertex has not either incoming or outgoing arcs then we shall call it the mute node. This naming being adopted we may say now: -graded poset may be identified with -ary relation as above iff it is -graded poset with no mute nodes. Equivalently - zero columns or rows in biadjacency matrices are forbidden. See and compare figures below.

1.4. Examples of

Let denotes arbitrary natural numbers valued sequence. Let be the Hasse matrix i.e. adjacency matrix of cover relation digraph denominated by sequence [1]. Then the zeta matrix for the denominated by cobweb poset is of the form [1] (see also [15-20,4]):

Example.1 . The incidence matrix for the N-cobweb poset.

Note that the matrix representing uniquely its corresponding cobweb poset does exhibits a staircase structure of zeros above the diagonal (see above, see below) which is characteristic to Hasse diagrams of all cobweb posets and for graded posets it is characteristic too.

Example.2 . The matrix for the Fibonacci cobweb poset associated to -KoDAG Hasse digraph.

The above remarks are visualized as below [15-20,4]. Namely - apart from - label, the another label and simultaneously visual code of cobweb graded poset is its ”‘La scala”’ descending down there to infinity with picture which looks like that below.

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 0 0 0 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 0 0 0 0 - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 0 0 0 - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 0 0 - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 0 - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - -

1 0 0 0 0 0 0 0 0 0 0 0 - - - - - - -

1 0 0 0 0 0 0 0 0 0 0 - - - - - - -

1 0 0 0 0 0 0 0 0 0 - - - - - - -

1 0 0 0 0 0 0 0 0 - - - - - - -

1 0 0 0 0 0 0 0 - - - - - - -

1 0 0 0 0 0 0 - - - - - - -

1 0 0 0 0 0 - - - - - - -

1 0 0 0 0 - - - - - - -

1 0 0 0 - - - - - - -

1 0 0 - - - - - - -

1 0 - - - - - - -

1 - - - - - - -

…………………….

and so on

Example.3 La Scala di Fibonacci . The staircase structure of incidence matrix for =Fibonacci sequence

Note. The picture above is drawn for the sequence , where are Fibonacci numbers.

Description of the Figure ”‘La Scala di Fibonacci”’ following [15-20,4]. If one defines (see: [15-20] and for earlier references therein as well as in all [1-8]) the Fibonacci poset with help of its incidence matrix representing uniquely then one arrives at with easily recognizable staircase-like structure - of zeros in the upper part of this upper triangle matrix . This structure is depicted by the Figure ”La Scala di Fibonacci”’ where: empty places mean zero values (under diagonal) and filled with – places mean values one (above the diagonal).

Advice. Simultaneous perpetual Exercises. How the all above and coming figures , formulas and expressions change (simplify) in the case of replacing the ring of integers in .

Comment 4. The given -denominated staircase zeros structure above the diagonal of zeta matrix is the unique characteristics of its corresponding -KoDAG Hasse digraphs, where denotes any natural numbers valued sequence as shown below.

For that to deliver we use the Gaussian coefficients inherited upside down notation i.e. (see [1-16], [27-30],and the Appendix in [9] extracted from [32]) and recall the Upside Down Notation Principle.

Let us also easier the portraying task putting . Then - apart from - label, the another label and simultaneously visual code of cobweb graded poset is its ”‘La scala”’ descending down there to infinity with picture which looks like that below , where

recall the is an arbitrary natural numbers valued sequence finite or infinite as .

1 ( - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

……………………………………………………………………………………………………………. 0 … 0 1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0 … 0 0 1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 … 0 0 0 1 ( - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

………………………………………………………………………………………………………………

0 … 0 0 0 1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 … 0 0 0 0 1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 … 0 0 0 0 0 1 ( - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

…………………………………………………………………………………………………………….. 0 …0 0 0 …0 1 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0 …0 0 0 …0 0 1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 …0 0 0 …0 0 0 1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 …0 0 0 …0 0 0 0 1 ( - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

and so on

Example.4 La scala F-Generale. The assumptive, perspicacious staircase structure of the incidence matrix for any F natural numbers valued sequence

Another special case Example is delivered by the Fig.5 below.

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

……………………………………

and so on

……………………………………

1 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 - - - - - - - - - - - - - - - - - - - - - - - - - - - -

and so on

Example.5 . The matrix for ( and for ) the special sequence F constituting the label sequence denominating cobweb poset associated to -KoDAG Hasse digraph.

Advice. Simultaneous perpetual Exercises. How the all above and coming picture Examples, figures, formulas and expressions change (simplify) in the case of replacing the ring of integers in .

1.5. Graded Posets’ matrix formula.

Recall now following SNACK that any graded poset with the finite set of minimal elements is an - sequence denominated sub-poset of its corresponding cobweb poset.
The Observation 2 in SNACK supplies the simple recipe for the biadjacency (reduced adjacency) matrix of Hasse digraph coding any given graded poset with the finite set of minimal elements. The recipe for zeta matrix is then standard. We illustrate this by the SNACK source example; the source example as the adjacency matrices i.e zeta matrices of any given graded poset with the finite set of minimal elements are sub-matrices of their corresponding cobweb posets and as such have the same block matrix structure and differ ”‘only”’ by eventual additional zeros in upper triangle matrix part while staying to be of the same cobweb poset block type.

The explicit expression for zeta matrix of cobweb posets via known blocks of zeros and ones for arbitrary natural numbers valued - sequence was given in [1] due to more than mnemonic efficiency of the up-side-down notation being applied (see [1] and references therein). With this notation inspired by Gauss and replacing - natural numbers with ”” numbers (Note. The Upside Down Notation Principle has been used in [1]) one gets :

and

where stays for matrix of ones i.e. ; and

Particular examples of the above block structure of matrix (resulting from being a result of natural join operations on the way) are supplied by Examples 1 ,2,3,4,5 above and Examples 6, 7, 8 represented by Fig.4, Fig.5, Fig.6 below. As a matter of fact - all elements of the incidence algebra including i.e. characteristic function of the partial order or Möbius function (as exemplified with Examples 9,10,11,12 below) have the same block structure encoded by sequence chosen. Recall that from denotes commutative ring and for example might be taken to be Boolean algebra , the field the ring of integers or real or complex or -adic numbers.

Namely, arbitrary is of the form

where denotes diagonal matrix while stays for arbitrary matrix and both with matrix elements from the ring = , , etc.

In more detail: it is trivial to note that all elements - including for which - are of matrix block form resulting from of the subsequent bipartite digraphs i.e.

where denote corresponding matrices with matrix elements from the ring = , , etc. However… for some seemingly most useful of them …

The New Name: -natural .

In the case of or August Ferdinand Möbius matrices motivating examples of specifically natural elements (i.e. -natural including those obtained via the ruling formula) - so in the case of such type elements we ascertain - and may prove via just see it - that

where the rectangular ”‘zero-one”’ matrices from Observation 2. are obtained from the -cobweb poset matrices by replacing some ones by zeros.

Moreover (see Observation 3) - in the case of Möbius matrix as it is obligatory .

The motivating example of -natural element of the incidence algebra is due to the algorithm of the ruling formula considered over the ring in this particular case element:

where