do a groundbreaking proof in mereology

--

I’ll isolate a precise mereological problem, develop a theorem with explicit assumptions, test counterexamples, and assess its novelty against the literature.

Reviewed mereology discussions, axioms, open problems, and proof standards

Refined theorem ideas

Tested modal expressibility

Assessed definability limits

Selected a provable result

I’m pursuing two routes: expressibility of finite fusion over arbitrary relations, and a potentially stronger necessity result for the existence of a smallest fusion completion.

(http://www.autoadmit.com/thread.php?thread_id=5910619&forum_id=2#50177092)

I obtained a candidate original result about when a mereological structure has a smallest completion. It gives this exact classification:

An incomplete core mereology has a dense fusion completion that embeds into every other dense fusion completion, preserving fusion and both preserving and reflecting parthood, if and only if the original structure consists of two incomparable atoms.

Thus, for an incomplete structure with three or more objects, such a completion does not exist.

The qualification concerning the comparison maps is essential. When maps only preserve parthood, a weaker minimum can exist. I also obtained an exact criterion for that case.

The closest result I located is in Imanol Mozo Carollo’s How to take cats together: Theorems 7.3 and 7.5 establish a sufficient atomicity condition for the weaker minimum. The proposed contributions here are the converse, the classification for order embeddings, and the distinction concerning unique comparison maps. Priority remains unverified

--

rest is mostly latex. should we give the candidate proof to codex to finish?

(http://www.autoadmit.com/thread.php?thread_id=5910619&forum_id=2#50177140)

The terminology follows Imanol Mozo Carollo’s How to take cats together, whose Theorems 7.3 and 7.5 establish sufficiency of the atomicity condition for the weaker comparison maps. The proof below establishes the stated classification, proves the converse for those weaker maps, and determines what happens when comparison maps must be unique. Historical priority for these conclusions remains unverified.

Work in ZFC and let \(P\) be a nonempty partially ordered set, with parthood denoted by \(\leq\). For any poset \(Q\), objects \(x,y\in Q\) overlap when some \(z\in Q\) satisfies both \(z\leq x\) and \(z\leq y\), and write this relation as \(x\circ_Q y\). For \(A\subseteq Q\), define

\[

O_Q(A)=\{y\in Q:\text{there is }a\in A\text{ such that }y\circ_Q a\},

\]

and abbreviate \(O_Q(\{x\})\) by \(O_Q(x)\). An object \(x\) is a fusion of a nonempty \(A\) when \(O_Q(x)=O_Q(A)\), and \(Q\) is complete for fusion when every nonempty subset has a fusion.

An atom of \(P\) is an object with no proper part in \(P\), and a subset closed under taking smaller objects will be called a lower set. Write \(\operatorname{At}(P)\) for the set of atoms of \(P\). A poset is atomic when every object has an atom below it. All uses of completeness concern arbitrary nonempty subsets, and incompleteness means that at least one such subset lacks a fusion.

A dense fusion completion can be represented by a poset \(M\) containing \(P\), with exactly the original order on \(P\), such that \(M\) is complete for fusion and every \(m\in M\) has some \(p\in P\) below it. The density arguments below show that these conditions also preserve and reflect original overlap and fusion, and make every object of \(M\) a fusion of original objects. Thus this description satisfies the additional requirements in the cited definition of a dense fusion completion.

A comparison map \(g\) between two completions must fix every element of \(P\). More generally, if the designated embeddings of \(P\) are \(i_C\) and \(i_M\), this requirement is

\[

g(i_C(p))=i_M(p)\qquad(p\in P).

\]

The map preserves fusion when it sends every fusion \(x\) of a nonempty set \(X\) to a fusion of \(g[X]\). An order embedding preserves and reflects order, whereas a monotone injection is injective and preserves order without a requirement to reflect it.

Density determines fusion from original objects. Let \(M\) be any order extension with \(P\) dense in the stated sense. If two original objects overlap in \(P\), they overlap in \(M\) using their original common part. Conversely, if they share a part in \(M\), density supplies an original object below that part, so they already overlap in \(P\).

For \(m\in M\), define its original parts and its original overlap profile by

\[

S_M(m)=\{p\in P:p\leq m\},\qquad

R_M(m)=\{p\in P:p\circ_M m\}.

\]

The set \(S_M(m)\) is a nonempty lower set of \(P\). Moreover,

\[

R_M(m)=O_P(S_M(m)),

\]

since an original object overlapping \(m\) has, by density, an original common part with \(m\), while overlap with an original part of \(m\) immediately gives overlap with \(m\).

For a nonempty \(X\subseteq M\), the exact test for fusion is

\[

m\text{ is a fusion of }X

\quad\text{if and only if}\quad

R_M(m)=\bigcup_{x\in X}R_M(x).

\]

Necessity follows by restricting the definition of fusion to original objects. For sufficiency, if \(y\circ_M m\), choose an original \(p\) below a common part of \(y,m\); the profile equality makes \(p\) overlap some \(x\in X\), so \(y\) overlaps that \(x\). Conversely, if \(y\) overlaps some \(x\in X\), choose an original \(p\) below a common part of \(y,x\); the same equality makes \(p\) overlap \(m\), and hence \(y\) overlaps \(m\).

For original objects, the profile \(R_M(p)\) is exactly \(O_P(p)\), as overlap on \(P\) has already been shown to remain unchanged. Applying the fusion test therefore proves both preservation and reflection of every original fusion. Applying it to \(S_M(m)\) also proves that every \(m\in M\) is a fusion of the nonempty subset \(S_M(m)\subseteq P\).

Atoms of \(P\) remain atoms in every such dense extension. Indeed, if \(x\leq a\) and \(a\) is an original atom, choose \(p\in P\) with \(p\leq x\); then \(p\leq a\) forces \(p=a\), and antisymmetry forces \(x=a\). It follows that an original atom \(a\) overlaps an object \(m\in M\) exactly when \(a\leq m\).

A general completion construction supplies the required comparison targets. Let \(\mathcal U\) be the collection of profiles \(O_P(A)\), for nonempty \(A\subseteq P\), which differ from \(O_P(p)\) for every \(p\in P\). Call these the missing profiles and, for \(U\in\mathcal U\), define

\[

K(U)=\{p\in P:O_P(p)\subseteq U\}.

\]

If \(U=O_P(A)\), then \(A\subseteq K(U)\), the set \(K(U)\) is lower, and

\[

O_P(K(U))=U.

\]

To verify these properties, note first that \(x\leq p\) gives \(O_P(x)\subseteq O_P(p)\), so \(K(U)\) is lower. Each \(a\in A\) has \(O_P(a)\subseteq O_P(A)=U\), giving \(A\subseteq K(U)\) and hence \(U\subseteq O_P(K(U))\). Conversely, every \(p\in K(U)\) has \(O_P(p)\subseteq U\), and taking their union gives \(O_P(K(U))\subseteq U\).

For each \(U\in\mathcal U\), suppose a nonempty lower set \(E(U)\subseteq P\) is given with \(O_P(E(U))=U\). Adjoin one fresh object \(t_U\) for each missing profile, keeping the order on \(P\) and declaring that \(p<t_U\) holds exactly when \(p\in E(U)\). Distinct added objects are incomparable, and no added object lies below an original object. Denote the resulting structure by \(C_E\).

This relation is a partial order: reflexivity is included, antisymmetry follows from the absence of reverse relations involving added objects, and transitivity involving an added object follows because \(E(U)\) is lower. Each added object has an original part since \(E(U)\) is nonempty, so \(P\) is dense in \(C_E\). The density calculation gives

\[

R_{C_E}(p)=O_P(p),\qquad R_{C_E}(t_U)=U.

\]

To verify completeness, take any nonempty \(X\subseteq C_E\) and put

\[

B=\bigcup_{x\in X}S_{C_E}(x).

\]

This is a nonempty subset of \(P\), and the profile calculation gives

\[

\bigcup_{x\in X}R_{C_E}(x)=O_P(B).

\]

If this profile equals \(O_P(p)\) for an original \(p\), that \(p\) has the required profile, while otherwise the added object \(t_{O_P(B)}\) has it. The fusion test proves that the selected object is a fusion of \(X\), verifying that \(C_E\) is a dense fusion completion.

In this construction each missing profile \(U\) has exactly one representative, namely \(t_U\). Taking \(E(U)=K(U)\) for every missing profile gives a particular completion denoted by \(G\), corresponding to Mozo Carollo’s \(G(P)\). The preceding argument verifies its completion properties directly.

Any completion with the asserted embedding property must be isomorphic to \(G\) over \(P\). Suppose \(L\) has that property and choose its comparison embedding \(j\) into \(G\). Given \(U=O_P(A)\in\mathcal U\), choose in \(L\) a fusion \(x\) of \(A\); preservation of fusion makes \(j(x)\) a fusion of \(A\) in \(G\), so the unique representative property forces \(j(x)=t_U\). Thus \(j\) reaches every added object and also every original object, since it fixes \(P\). A surjective order embedding is an order isomorphism, which also preserves fusion in both directions, so \(G\) itself must admit the required comparison embedding into every dense completion.

We next prove that every \(K(U)\) must consist entirely of atoms. Suppose, toward a contradiction, that \(p\in K(U)\) has a proper part \(q\in P\). Define

\[

E(U)=\{x\in K(U):p\nleq x\}.

\]

This is a lower set containing \(q\), and every \(x\in K(U)\) has an element of \(E(U)\) below it: use \(x\) when \(p\nleq x\), and use \(q\) when \(p\leq x\).

It follows that \(O_P(E(U))=O_P(K(U))=U\). For the nontrivial inclusion, if \(z\) overlaps \(x\in K(U)\), choose a common part \(w\leq z,x\), which belongs to \(K(U)\), and then choose an element of \(E(U)\) below \(w\). This element witnesses that \(z\) overlaps \(E(U)\), while the other inclusion follows from \(E(U)\subseteq K(U)\).

Retain \(E(V)=K(V)\) for every other missing profile \(V\). The general construction produces a dense completion in which \(p\nleq t_U\), although \(p\leq t_U\) in \(G\). Every comparison map preserving fusion and fixing \(P\) must send the representative \(t_U\) in \(G\) to the unique representative of \(U\) in this target. It cannot preserve the relation \(p\leq t_U\), giving the contradiction and proving

\[

K(U)\subseteq\operatorname{At}(P)\qquad(U\in\mathcal U).

\]

Now let \(\mathcal D^+(P)\) consist of all nonempty lower subsets of \(P\), ordered by inclusion, and represent each \(p\in P\) by \(J(p)=\{x\in P:x\leq p\}\). The map \(J\) preserves and reflects order, and its image is dense since every nonempty lower set contains some \(p\) and hence contains \(J(p)\). Two members of \(\mathcal D^+(P)\) overlap exactly when their intersection is nonempty: an element of the intersection supplies a common principal lower set, and every common lower bound is itself a nonempty subset of the intersection. Consequently, the union of any nonempty family is a fusion of that family, so \(\mathcal D^+(P)\) is a dense fusion completion.

Let \(h\) be the required order embedding of \(G\) into \(\mathcal D^+(P)\), with \(h(p)=J(p)\) for original objects. For every \(p\in P\), membership \(p\in h(t_U)\) is equivalent to \(J(p)\subseteq h(t_U)\), since \(h(t_U)\) is lower. Preservation and reflection of order make this equivalent to \(p\leq t_U\) in \(G\), which is equivalent to \(p\in K(U)\). Therefore

\[

h(t_U)=K(U)\qquad(U\in\mathcal U).

\]

Suppose \(u\in P\) were a fusion of all of \(P\). Every original atom would overlap \(u\) and therefore lie below it, making \(K(U)\subseteq J(u)\) for every missing profile \(U\). There is at least one such \(U\), since \(P\) is incomplete. The equality \(h(t_U)=K(U)\) would then give \(h(t_U)\subseteq h(u)\), forcing \(t_U\leq u\) by order reflection, contrary to the construction of \(G\).

Thus \(P\) has no fusion of all its objects, and the profile \(O_P(P)=P\) belongs to \(\mathcal U\). For that profile \(K(P)=P\), so the atomic conclusion already proved says that every object of \(P\) is an atom. Consequently, \(P\) is an antichain.

If this antichain contained at least three objects, choose subsets \(B\subsetneq D\subseteq P\) with \(|B|=2\) and \(|D|=3\). In an antichain overlap is equality, so \(O_P(B)=B\) and \(O_P(D)=D\), and both profiles are missing. The objects \(t_B,t_D\) are incomparable in \(G\), but their forced images under \(h\) are \(B,D\), which are comparable by inclusion. This contradicts order reflection, proving that \(|P|\leq2\).

A nonempty poset with one object is complete for fusion, so incompleteness leaves exactly two incomparable atoms \(a,b\). To prove the converse, form \(L=P\cup\{t\}\) with \(a<t\) and \(b<t\), which is the general construction for this antichain’s single missing profile. In an arbitrary dense fusion completion \(M\), choose a fusion \(m\) of \(\{a,b\}\); the atom lemma gives \(a\leq m\) and \(b\leq m\), and \(m\) differs from both atoms because its original overlap profile is \(\{a,b\}\). Sending \(t\) to \(m\) and fixing \(a,b\) therefore preserves and reflects every order relation among the three objects.

This map preserves the original overlap profile of each of its three arguments. Taking unions shows that it preserves the profile of every nonempty subset of \(L\). The fusion test therefore proves that it preserves every fusion, completing the proof of the main classification.

For the weaker comparison maps, the exact criterion is local atomicity at every missing fusion. More precisely, a dense completion admitting an injective map preserving order and fusion over \(P\) into every other dense completion exists if and only if the following condition holds: whenever nonempty \(A\subseteq P\) has no fusion in \(P\), every \(x\leq a\) with \(a\in A\) has an atom of \(P\) below it. Equivalently, the lower set generated by each such \(A\) is atomic.

For necessity, let \(L\) have this weaker property, take a subset \(A\) lacking a fusion in \(P\), and choose a fusion \(c\) of \(A\) in \(L\). Put \(U=O_P(A)\), which is a missing profile. Suppose an original nonatom \(p\) satisfies \(p\leq c\), so \(O_P(p)\subseteq R_L(c)=U\) and therefore \(p\in K(U)\).

The deletion construction above gives a dense completion \(C_E\) whose unique representative \(t_U\) of this profile satisfies \(p\nleq t_U\). A comparison map \(g\) preserving fusion must send \(c\) to \(t_U\), and order preservation must give \(g(p)\leq g(c)\). Since \(g(p)=p\), this is a contradiction, showing that every original part of \(c\) is an atom.

Take any \(x\leq a\) with \(a\in A\). The object \(x\) overlaps \(A\), so the fusion condition makes it overlap \(c\) in \(L\). Density supplies an original object below a common part of \(x,c\), and that original object is an atom by the preceding paragraph. It is an atom below \(x\), proving necessity of the stated condition.

For sufficiency, assume the condition and define

\[

E(U)=\operatorname{At}(P)\cap K(U)\qquad(U\in\mathcal U).

\]

Fix a representation \(U=O_P(A)\) and take \(z\in U\), so some \(a\in A\) and some common part \(w\leq z,a\) exist in \(P\). The condition supplies an atom \(b\leq w\), and then \(b\leq a\) gives \(O_P(b)\subseteq U\), so \(b\in E(U)\) and \(z\) overlaps \(E(U)\). Thus \(O_P(E(U))=U\), since the reverse inclusion follows from \(E(U)\subseteq K(U)\), and \(E(U)\) is a nonempty lower set consisting of atoms.

Apply the general construction with these sets and call the resulting dense completion \(H\), corresponding to Mozo Carollo’s \(H(P)\). Given any dense completion \(M\), choose for each missing \(U\) a fusion \(m_U\) of \(E(U)\) in \(M\), using the axiom of choice for this family. Define \(g\) to fix \(P\) and satisfy \(g(t_U)=m_U\), so

\[

R_M(g(z))=R_H(z)\qquad(z\in H).

\]

Distinct missing profiles force distinct chosen objects \(m_U\), and none is original because no original object has a missing profile. Thus \(g\) is injective, and it preserves the order on \(P\). Each additional strict relation in \(H\) has the form \(a<t_U\) with \(a\in E(U)\) an original atom; since \(m_U\) fuses \(E(U)\), it overlaps \(a\), and the atom lemma gives \(a\leq m_U\). Hence \(g\) preserves every order relation.

If \(z\) is a fusion of nonempty \(X\subseteq H\), the fusion test expresses that fact as equality between its original profile and the union of the profiles of the members of \(X\). The displayed profile identity for \(g\) transfers that equality to \(M\). Applying the same test in \(M\) proves that \(g(z)\) is a fusion of \(g[X]\), establishing sufficiency and completing the proof of the weaker criterion.

Uniqueness of every comparison map fails for every incomplete \(P\). Form a target \(T\) by replacing each added object \(t_U\) of \(G\) with two incomparable objects \(t_U^0,t_U^1\), each having original parts exactly \(K(U)\). Keep distinct added objects incomparable and retain the order on \(P\). The general construction’s order and density verification still applies, and the completeness proof still works because every required missing profile has representatives.

The permutation \(\sigma\) fixing \(P\) and swapping \(t_U^0\) with \(t_U^1\) for every \(U\) is an order automorphism of \(T\), so it preserves fusion. Suppose a completion \(L\) admitted exactly one allowed comparison map into every dense completion, and let \(g\) be its map into \(T\). Choose a subset \(A\subseteq P\) lacking a fusion in \(P\) and a fusion \(c\) of \(A\) in \(L\); then \(g(c)\) must be one of the two added representatives of \(O_P(A)\). The map \(\sigma\circ g\) is another allowed comparison map, whether the chosen class consists of order embeddings or monotone injections preserving fusion, and it differs from \(g\) at \(c\), contradicting uniqueness.

For completeness, if \(P\) already has every required fusion, take \(P\) itself as its completion. Every map over \(P\) from this source is forced to be the designated embedding into the target, which preserves order and fusion by the density argument. Thus requiring unique comparison maps is possible exactly when the original structure is already complete.

(http://www.autoadmit.com/thread.php?thread_id=5910619&forum_id=2#50177199)

Write \(O(A)\) for the objects of \(P\) overlapping some member of \(A\), where overlap means sharing a part. Call \(U=O(A)\) a *missing profile* when \(A\) is nonempty and no original object has overlap profile \(U\). Define \(K_U=\{p\in P:O(\{p\})\subseteq U\}\).

Mozo Carollo’s construction \(G(P)\) adds one object \(f_U\) for each missing profile, with original parts exactly \(K_U\). Added objects are mutually incomparable and lie below no original object. Theorem 6.5 establishes this completion, whose density follows directly from the construction. [arxiv.org](https://arxiv.org/pdf/2410.02350?utm_source=chatgpt.com)

Suppose a completion \(C\) embeds into every dense completion through the required comparison maps. Its embedding into \(G(P)\) must reach every \(f_U\), since \(C\) must supply each missing fusion and \(f_U\) is its only possible image. The embedding already fixes every original object, so it is surjective, making **\(C\) isomorphic to \(G(P)\)** over \(P\).

In a dense extension, fusion can be checked using overlap with original objects alone. Every overlap has an original object below its common part, so a failure of the fusion condition would have an original witness. Consequently, the construction remains a dense fusion completion if \(K_U\) is replaced by any downward closed subset \(E_U\) satisfying \(O(E_U)=U\). Arbitrary unions of the represented profiles are again profiles of subsets of \(P\), ensuring completeness.

Now suppose some \(p\in K_U\) has a proper part \(q\) in \(P\). Replace the original parts beneath \(f_U\) by

\(E_U=\{x\in K_U:p\nleq x\}\).

Every member of \(K_U\) has a member of \(E_U\) below it, since either that member itself remains or \(q\) lies below it, giving \(O(E_U)=U\). The resulting structure is therefore another dense fusion completion, but its unique representative of \(U\) no longer has \(p\) as a part. Any comparison map from \(G(P)\) must send \(f_U\) to that representative, violating order preservation, so **every \(K_U\) must consist entirely of atoms**.

Consider another completion, \(\mathcal D^+(P)\), consisting of all nonempty downward closed subsets of \(P\), ordered by inclusion, with \(p\) represented by \(P_{\le p}=\{x\in P:x\le p\}\). Unions provide fusions, and the principal subsets \(P_{\le p}\) provide density. An order embedding of \(G(P)\) into this completion must send \(f_U\) to exactly \(K_U\), since preserving and reflecting its relations with original objects determines every member of its image.

If \(P\) already had a fusion \(u\) of all its objects, every atom would lie below \(u\), since an atom that overlaps \(u\) is a part of \(u\). Consequently, \(K_U\subseteq P_{\le u}\) for every missing profile \(U\). This makes the image of \(f_U\) a part of the image of \(u\), although \(f_U\nleq u\) in \(G(P)\), violating order reflection. Thus \(P\) itself must be a missing profile.

Since \(K_P=P\), every original object must therefore be an atom. If \(P\) contained at least three objects, choose \(B\subsetneq D\subseteq P\) with two and three members respectively, making both missing profiles. Their representatives in \(G(P)\) are incomparable, but their forced images \(B\) and \(D\) in \(\mathcal D^+(P)\) are comparable. Order reflection fails again, so an incomplete \(P\) must have exactly two objects.

Conversely, for two incomparable atoms \(a,b\), take the completion obtained by adjoining \(t\) with \(a<t\) and \(b<t\). In any dense completion, \(a,b\) remain atoms, so any fusion \(v\) of them necessarily lies above both. Sending \(t\) to \(v\) and fixing \(a,b\) preserves fusion and both preserves and reflects parthood, establishing sufficiency.

**The exact criterion for the weaker comparison maps is also provable:** every object below a member of a subset lacking a fusion must have an atom below it. For injective maps preserving fusion and order, this is the sufficient hypothesis of Mozo Carollo’s Theorems 7.3 and 7.5. The deletion construction above supplies the converse. [arxiv.org](https://arxiv.org/pdf/2410.02350?utm_source=chatgpt.com)

For necessity, let \(C\) be a proposed minimum under those weaker maps, and let \(c\) fuse a subset \(A\) lacking a fusion in \(P\). If an original nonatom \(p\) lies below \(c\), the deletion construction supplies a target whose unique fusion of \(A\) is not above \(p\), contradicting order preservation. Thus every original object below \(c\) is an atom. For any original \(x\le a\) with \(a\in A\), fusion gives overlap between \(x\) and \(c\), and density supplies an original atom below both.

Requiring a **unique comparison map into every dense completion** is impossible for any incomplete \(P\). Replace every added \(f_U\) in \(G(P)\) by two incomparable copies with identical original parts, obtaining another dense completion with an automorphism fixing \(P\) and swapping the copies. Any proposed universal source must send a fusion of a missing subset to an added copy, and composing that comparison map with the automorphism produces a second distinct comparison map.

(http://www.autoadmit.com/thread.php?thread_id=5910619&forum_id=2#50177173)