{
  "index": "1957-B-5",
  "type": "COMB",
  "tag": [
    "COMB",
    "ALG"
  ],
  "difficulty": "",
  "question": "5. With each subset \\( X \\) of a set is associated a second subset \\( f(X) \\). The association is such that whenever \\( X \\) contains \\( Y \\) then \\( f(X) \\) contains \\( f(Y) \\). Show that for some set \\( A, f(A)=A \\).",
  "solution": "Solution. Let the given set be \\( S \\). Define\n\\[\n\\mathfrak{C}=\\{X \\subseteq S: X \\subseteq f(X)\\}\n\\]\nand let \\( A \\) be the union of all members of \\( \\mathcal{C} \\). We shall prove that \\( A=f(A) \\).\nSuppose \\( X \\in \\mathbb{C} \\); then \\( X \\subseteq f(X) \\) and \\( X \\subseteq A \\). Therefore, \\( f(X) \\subseteq f(A) \\), by hypothesis, so \\( X \\subseteq f(A) \\). By the definition of union\n\\[\nA \\subseteq f(A) .\n\\]\n\nBy the hypothesis, \\( f(A) \\subseteq f(f(A)) \\), so \\( f(A) \\in \\mathbb{C} \\). Again by definition of union,\n\\[\nf(A) \\subseteq A .\n\\]\n\nComparing (1) and (2), we see that \\( A=f(A) \\), as claimed.\nRemarks. By essentially the same argument one can prove the KnasterTarski fixed point theorem-namely: Every order-preserving mapping of a complete lattice into itself has a fixed element. See G. Szasz, Introduction to Lattice Theory, Academic Press, New York, 1963.\n\nFraenkel (in Abstract Set Theory, North Holland Publishing Co., 1953) ascribes the result to Dedekind, whose work however was not published until 1932, and independently to Peano and Zermelo.",
  "vars": [
    "X",
    "Y",
    "A"
  ],
  "params": [
    "f",
    "S",
    "C"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "X": "subsetx",
        "Y": "subsety",
        "A": "fixset",
        "f": "function",
        "S": "originalset",
        "C": "collectn"
      },
      "question": "Problem:\n<<<\n5. With each subset \\( subsetx \\) of a set is associated a second subset \\( function(subsetx) \\). The association is such that whenever \\( subsetx \\) contains \\( subsety \\) then \\( function(subsetx) \\) contains \\( function(subsety) \\). Show that for some set \\( fixset, function(fixset)=fixset \\).\n>>>\n",
      "solution": "Solution:\n<<<\nSolution. Let the given set be \\( originalset \\). Define\n\\[\ncollectn=\\{subsetx \\subseteq originalset: subsetx \\subseteq function(subsetx)\\}\n\\]\nand let \\( fixset \\) be the union of all members of \\( collectn \\). We shall prove that \\( fixset=function(fixset) \\).\nSuppose \\( subsetx \\in collectn \\); then \\( subsetx \\subseteq function(subsetx) \\) and \\( subsetx \\subseteq fixset \\). Therefore, \\( function(subsetx) \\subseteq function(fixset) \\), by hypothesis, so \\( subsetx \\subseteq function(fixset) \\). By the definition of union\n\\[\nfixset \\subseteq function(fixset) .\n\\]\n\nBy the hypothesis, \\( function(fixset) \\subseteq function(function(fixset)) \\), so \\( function(fixset) \\in collectn \\). Again by definition of union,\n\\[\nfunction(fixset) \\subseteq fixset .\n\\]\n\nComparing (1) and (2), we see that \\( fixset=function(fixset) \\), as claimed.\nRemarks. By essentially the same argument one can prove the KnasterTarski fixed point theorem-namely: Every order-preserving mapping of a complete lattice into itself has a fixed element. See G. Szasz, Introduction to Lattice Theory, Academic Press, New York, 1963.\n\nFraenkel (in Abstract Set Theory, North Holland Publishing Co., 1953) ascribes the result to Dedekind, whose work however was not published until 1932, and independently to Peano and Zermelo.\n>>>\n"
    },
    "descriptive_long_confusing": {
      "map": {
        "X": "sandalwood",
        "Y": "lighthouse",
        "A": "clementine",
        "f": "dandelion",
        "S": "tablespoon",
        "C": "raincloud"
      },
      "question": "5. With each subset \\( sandalwood \\) of a set is associated a second subset \\( dandelion(sandalwood) \\). The association is such that whenever \\( sandalwood \\) contains \\( lighthouse \\) then \\( dandelion(sandalwood) \\) contains \\( dandelion(lighthouse) \\). Show that for some set \\( clementine, dandelion(clementine)=clementine \\).",
      "solution": "Solution. Let the given set be \\( tablespoon \\). Define\n\\[\n\\mathfrak{raincloud}=\\{sandalwood \\subseteq tablespoon: sandalwood \\subseteq dandelion(sandalwood)\\}\n\\]\nand let \\( clementine \\) be the union of all members of \\( \\mathcal{raincloud} \\). We shall prove that \\( clementine=dandelion(clementine) \\).\nSuppose \\( sandalwood \\in \\mathbb{raincloud} \\); then \\( sandalwood \\subseteq dandelion(sandalwood) \\) and \\( sandalwood \\subseteq clementine \\). Therefore, \\( dandelion(sandalwood) \\subseteq dandelion(clementine) \\), by hypothesis, so \\( sandalwood \\subseteq dandelion(clementine) \\). By the definition of union\n\\[\nclementine \\subseteq dandelion(clementine) .\n\\]\n\nBy the hypothesis, \\( dandelion(clementine) \\subseteq dandelion(dandelion(clementine)) \\), so \\( dandelion(clementine) \\in \\mathbb{raincloud} \\). Again by definition of union,\n\\[\ndandelion(clementine) \\subseteq clementine .\n\\]\n\nComparing (1) and (2), we see that \\( clementine=dandelion(clementine) \\), as claimed.\nRemarks. By essentially the same argument one can prove the KnasterTarski fixed point theorem-namely: Every order-preserving mapping of a complete lattice into itself has a fixed element. See G. Szasz, Introduction to Lattice Theory, Academic Press, New York, 1963.\n\nFraenkel (in Abstract Set Theory, North Holland Publishing Co., 1953) ascribes the result to Dedekind, whose work however was not published until 1932, and independently to Peano and Zermelo."
    },
    "descriptive_long_misleading": {
      "map": {
        "X": "knownvalue",
        "Y": "fixeditem",
        "A": "shiftingset",
        "f": "staticrule",
        "S": "emptiness",
        "C": "chaosgroup"
      },
      "question": "5. With each subset \\( knownvalue \\) of a set is associated a second subset \\( staticrule(knownvalue) \\). The association is such that whenever \\( knownvalue \\) contains \\( fixeditem \\) then \\( staticrule(knownvalue) \\) contains \\( staticrule(fixeditem) \\). Show that for some set \\( shiftingset, staticrule(shiftingset)=shiftingset \\).",
      "solution": "Solution. Let the given set be \\( emptiness \\). Define\n\\[\n\\mathfrak{chaosgroup}=\\{knownvalue \\subseteq emptiness: knownvalue \\subseteq staticrule(knownvalue)\\}\n\\]\nand let \\( shiftingset \\) be the union of all members of \\( \\mathcal{chaosgroup} \\). We shall prove that \\( shiftingset=staticrule(shiftingset) \\).\nSuppose \\( knownvalue \\in \\mathbb{chaosgroup} \\); then \\( knownvalue \\subseteq staticrule(knownvalue) \\) and \\( knownvalue \\subseteq shiftingset \\). Therefore, \\( staticrule(knownvalue) \\subseteq staticrule(shiftingset) \\), by hypothesis, so \\( knownvalue \\subseteq staticrule(shiftingset) \\). By the definition of union\n\\[\nshiftingset \\subseteq staticrule(shiftingset) .\n\\]\n\nBy the hypothesis, \\( staticrule(shiftingset) \\subseteq staticrule(staticrule(shiftingset)) \\), so \\( staticrule(shiftingset) \\in \\mathbb{chaosgroup} \\). Again by definition of union,\n\\[\nstaticrule(shiftingset) \\subseteq shiftingset .\n\\]\n\nComparing (1) and (2), we see that \\( shiftingset=staticrule(shiftingset) \\), as claimed.\nRemarks. By essentially the same argument one can prove the KnasterTarski fixed point theorem-namely: Every order-preserving mapping of a complete lattice into itself has a fixed element. See G. Szasz, Introduction to Lattice Theory, Academic Press, New York, 1963.\n\nFraenkel (in Abstract Set Theory, North Holland Publishing Co., 1953) ascribes the result to Dedekind, whose work however was not published until 1932, and independently to Peano and Zermelo."
    },
    "garbled_string": {
      "map": {
        "X": "qzxwvtnp",
        "Y": "hjgrksla",
        "A": "vmlpqzrt",
        "f": "bwxsnmle",
        "S": "kdfhjprs",
        "C": "sgnvclta"
      },
      "question": "<<<\n5. With each subset \\( qzxwvtnp \\) of a set is associated a second subset \\( bwxsnmle(qzxwvtnp) \\). The association is such that whenever \\( qzxwvtnp \\) contains \\( hjgrksla \\) then \\( bwxsnmle(qzxwvtnp) \\) contains \\( bwxsnmle(hjgrksla) \\). Show that for some set \\( vmlpqzrt, bwxsnmle(vmlpqzrt)=vmlpqzrt \\).\n>>>",
      "solution": "<<<\nSolution. Let the given set be \\( kdfhjprs \\). Define\n\\[\n\\mathfrak{sgnvclta}=\\{qzxwvtnp \\subseteq kdfhjprs: qzxwvtnp \\subseteq bwxsnmle(qzxwvtnp)\\}\n\\]\nand let \\( vmlpqzrt \\) be the union of all members of \\( \\mathcal{sgnvclta} \\). We shall prove that \\( vmlpqzrt=bwxsnmle(vmlpqzrt) \\).\nSuppose \\( qzxwvtnp \\in \\mathbb{sgnvclta} \\); then \\( qzxwvtnp \\subseteq bwxsnmle(qzxwvtnp) \\) and \\( qzxwvtnp \\subseteq vmlpqzrt \\). Therefore, \\( bwxsnmle(qzxwvtnp) \\subseteq bwxsnmle(vmlpqzrt) \\), by hypothesis, so \\( qzxwvtnp \\subseteq bwxsnmle(vmlpqzrt) \\). By the definition of union\n\\[\nvmlpqzrt \\subseteq bwxsnmle(vmlpqzrt) .\n\\]\n\nBy the hypothesis, \\( bwxsnmle(vmlpqzrt) \\subseteq bwxsnmle(bwxsnmle(vmlpqzrt)) \\), so \\( bwxsnmle(vmlpqzrt) \\in \\mathbb{sgnvclta} \\). Again by definition of union,\n\\[\nbwxsnmle(vmlpqzrt) \\subseteq vmlpqzrt .\n\\]\n\nComparing (1) and (2), we see that \\( vmlpqzrt=bwxsnmle(vmlpqzrt) \\), as claimed.\nRemarks. By essentially the same argument one can prove the KnasterTarski fixed point theorem-namely: Every order-preserving mapping of a complete lattice into itself has a fixed element. See G. Szasz, Introduction to Lattice Theory, Academic Press, New York, 1963.\n\nFraenkel (in Abstract Set Theory, North Holland Publishing Co., 1953) ascribes the result to Dedekind, whose work however was not published until 1932, and independently to Peano and Zermelo.\n>>>"
    },
    "kernel_variant": {
      "question": "Let $G$ be an arbitrary (possibly infinite) group and write \n\n\\[\n\\operatorname{Sub}(G)=\\{\\,H\\le G\\,\\},\n\\]\n\nthe complete lattice of all subgroups of $G$, ordered by inclusion.  \nFix a non–empty index-set $\\Lambda$ and denote by $\\Lambda^{<\\omega}$ the set of all finite words  \n$\\sigma=(\\lambda_{1},\\dots,\\lambda_{n})$ (the empty word $\\varnothing$ is allowed).\n\nFor every $\\sigma\\in\\Lambda^{<\\omega}$ an order–preserving \\emph{complete} lattice endomorphism  \n\n\\[\n\\Phi_{\\sigma}\\colon\\operatorname{Sub}(G)\\longrightarrow\\operatorname{Sub}(G)\n\\]\n\nis given, i.e. $\\Phi_{\\sigma}$ preserves \\emph{all} joins and meets that exist in the lattice.  \nThese maps satisfy the structural requirement  \n\n\\[\n\\text{(C)\\quad Commutativity}\\qquad   \n\\Phi_{\\sigma}\\circ\\Phi_{\\tau}=\\Phi_{\\tau}\\circ\\Phi_{\\sigma}\\qquad\n\\text{for all finite words }\\sigma,\\tau.\n\\]\n\nA subgroup $H\\le G$ is called \\emph{$\\Phi$–stable} if $\\Phi_{\\sigma}(H)=H$ for every finite word $\\sigma$.\n\n(a) Prove that at least one $\\Phi$–stable subgroup of $G$ exists.\n\n(b) Put  \n\n\\[\n\\text{Pre}:=\\bigl\\{\\,H\\le G:\\; H\\le \\Phi_{\\sigma}(H)\\ \\text{for every } \\sigma \\bigr\\},\\qquad\n\\text{Post}:=\\bigl\\{\\,H\\le G:\\; \\Phi_{\\sigma}(H)\\le H\\ \\text{for every } \\sigma \\bigr\\},\n\\]\n\nand define  \n\n\\[\nB:=\\bigcap\\text{Post},\\qquad\nC:=\\left\\langle\\bigcup\\text{Pre}\\right\\rangle .\n\\]\n\nProve that $B$ is the least and $C$ the greatest $\\Phi$–stable subgroup of $G$.\n\n(c) Let $\\kappa:=|\\operatorname{Sub}(G)|$.  \nShow that $B$ and $C$ are obtained in fewer than $\\kappa^{+}$ transfinite steps by the monotone constructions  \n\nLeast fixed point  \n\n\\[\nB_{0}=\\{1\\},\\qquad\nB_{\\alpha+1}= \\bigvee_{\\sigma\\in\\Lambda^{<\\omega}}\\Phi_{\\sigma}(B_{\\alpha}),\\qquad\nB_{\\lambda}= \\bigvee_{\\beta<\\lambda}B_{\\beta}\\quad(\\lambda\\text{ limit});\n\\]\n\nGreatest fixed point  \n\n\\[\nC_{0}=G,\\qquad\nC_{\\alpha+1}= \\bigcap_{\\sigma\\in\\Lambda^{<\\omega}}\\Phi_{\\sigma}(C_{\\alpha}),\\qquad\nC_{\\lambda}= \\bigcap_{\\beta<\\lambda}C_{\\beta}\\quad(\\lambda\\text{ limit}).\n\\]\n\nProve that each transfinite sequence stabilises after fewer than $\\kappa^{+}$ steps and that its limit equals, respectively, $B$ and $C$.\n\n(The three parts together constitute a Knaster–Tarski fixed-point theorem for an arbitrary commuting family of complete lattice endomorphisms on $\\operatorname{Sub}(G)$.)\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Write $\\langle S\\rangle$ for the subgroup generated by a set $S$ and, for every family \n$\\mathcal H\\subseteq\\operatorname{Sub}(G)$, set  \n\n\\[\n\\bigvee\\mathcal H:=\\left\\langle\\bigcup_{H\\in\\mathcal H}H\\right\\rangle,\\qquad\n\\bigwedge\\mathcal H:=\\bigcap_{H\\in\\mathcal H}H .\n\\]\n\nStep 1.  Two auxiliary classes.  \n\n\\[\n\\text{Pre}:=\\{\\,H\\le G:H\\le\\Phi_{\\sigma}(H)\\ \\forall\\sigma\\,\\},\\qquad\n\\text{Post}:=\\{\\,H\\le G:\\Phi_{\\sigma}(H)\\le H\\ \\forall\\sigma\\,\\}.\n\\]\n\nBoth classes are non-empty: $\\{1\\}\\in\\text{Pre}$ and $G\\in\\text{Post}$.\n\n1a.  $\\text{Pre}$ is closed under arbitrary joins.  \nLet $\\mathcal H\\subseteq\\text{Pre}$ and put $K:=\\bigvee\\mathcal H=\\langle\\bigcup\\mathcal H\\rangle$.  \nFix $\\sigma$.  Each generator $g$ of $K$ lies in some $H\\in\\mathcal H$, hence  \n\n\\[\ng\\in H\\le\\Phi_{\\sigma}(H)\\subseteq\\Phi_{\\sigma}(K).\n\\]\n\nSince $\\Phi_{\\sigma}(K)$ is a subgroup, it contains every element of $K$, giving $K\\le\\Phi_{\\sigma}(K)$; thus $K\\in\\text{Pre}$.\n\n1b.  $\\text{Post}$ is closed under arbitrary meets.  \nLet $\\mathcal H\\subseteq\\text{Post}$ and set $L:=\\bigwedge\\mathcal H=\\bigcap\\mathcal H$.  \nFor each $\\sigma$,\n\n\\[\n\\Phi_{\\sigma}(L)=\\Phi_{\\sigma}\\!\\bigl(\\bigcap\\mathcal H\\bigr)\n               =\\bigcap_{H\\in\\mathcal H}\\Phi_{\\sigma}(H)\n               \\le\\bigcap_{H\\in\\mathcal H}H=L,\n\\]\n\nwhere the equality uses the fact that $\\Phi_{\\sigma}$ preserves arbitrary meets.  \nHence $L\\in\\text{Post}$.\n\nStep 2.  Extremal $\\Phi$–stable subgroups.  \nDefine  \n\n\\[\nB:=\\bigcap\\text{Post},\\qquad C:=\\bigvee\\text{Pre}.\n\\]\n\n2a.  $C$ is $\\Phi$–stable.  \nFix $\\sigma$.  Because every $H\\in\\text{Pre}$ satisfies $H\\le\\Phi_{\\sigma}(H)$, the join $C$ fulfils $C\\le\\Phi_{\\sigma}(C)$.  \nConversely, $\\Phi_{\\sigma}$ sends $\\text{Pre}$ into itself: if $H\\in\\text{Pre}$ and $\\tau$ is arbitrary then  \n\n\\[\n\\Phi_{\\tau}\\bigl(\\Phi_{\\sigma}(H)\\bigr)\n        =\\Phi_{\\sigma}\\bigl(\\Phi_{\\tau}(H)\\bigr)\n        \\ge\\Phi_{\\sigma}(H),\n\\]\n\nso $\\Phi_{\\sigma}(H)\\in\\text{Pre}$.  Join-closure of $\\text{Pre}$ yields $\\Phi_{\\sigma}(C)\\in\\text{Pre}$, whence $\\Phi_{\\sigma}(C)\\le C$.  Hence $\\Phi_{\\sigma}(C)=C$.\n\n2b.  $B$ is $\\Phi$–stable.  \nDually, $\\text{Post}$ is meet-closed and $\\Phi_{\\sigma}(\\text{Post})\\subseteq\\text{Post}$.  Therefore $\\Phi_{\\sigma}(B)\\in\\text{Post}$.  Because $B$ is the meet of $\\text{Post}$, we have  \n\n\\[\nB\\le\\Phi_{\\sigma}(B)\\le B,\n\\]\n\nso $\\Phi_{\\sigma}(B)=B$.\n\n2c.  Extremality.  Every $\\Phi$–stable subgroup lies in both $\\text{Pre}$ and $\\text{Post}$, hence $B\\le H\\le C$ for all $\\Phi$–stable $H$.\n\nParts (a) and (b) are complete.\n\nStep 3.  Transfinite construction of the least fixed point $B$.\n\nLet $\\kappa:=|\\operatorname{Sub}(G)|$ and define  \n\n\\[\nB_{0}=\\{1\\},\\qquad\nB_{\\alpha+1}= \\bigvee_{\\sigma\\in\\Lambda^{<\\omega}}\\Phi_{\\sigma}(B_{\\alpha}),\\qquad\nB_{\\lambda}= \\bigvee_{\\beta<\\lambda}B_{\\beta}\\quad(\\lambda\\text{ limit}).\n\\]\n\n3a.  Ascendingness.  \nBecause $B_{\\alpha}\\in\\text{Pre}$ (proved next) we have $B_{\\alpha}\\le\\Phi_{\\sigma}(B_{\\alpha})$ for every $\\sigma$, hence $B_{\\alpha}\\le B_{\\alpha+1}$.  Limits are joins, so the chain is ascending.\n\n3b.  $B_{\\alpha}\\in\\text{Pre}$ for all $\\alpha$ (transfinite induction).  \nBase $\\alpha=0$ is clear.  \nSuccessor: if $B_{\\alpha}\\in\\text{Pre}$ then each $\\Phi_{\\sigma}(B_{\\alpha})\\in\\text{Pre}$, and join-closure gives $B_{\\alpha+1}\\in\\text{Pre}$.  \nLimit: join-closure again.\n\n3c.  Stabilisation before stage $\\kappa^{+}$.  \nAn ascending chain of subgroups has length at most $\\kappa$, so some $\\alpha<\\kappa^{+}$ satisfies $B_{\\alpha}=B_{\\alpha+1}$.  Set $B_{*}:=B_{\\alpha}$.\n\n3d.  $\\Phi$–stability of the limit.  \nBy definition $B_{*}=\\bigvee_{\\sigma}\\Phi_{\\sigma}(B_{*})$, hence $\\Phi_{\\sigma}(B_{*})\\le B_{*}$ for every $\\sigma$.  Because $B_{*}\\in\\text{Pre}$ the reverse inclusion holds as well, yielding $\\Phi_{\\sigma}(B_{*})=B_{*}$.\n\n3e.  Minimality.  \nLet $H$ be $\\Phi$–stable.  By induction on $\\alpha$ and monotonicity of each $\\Phi_{\\sigma}$ we get $B_{\\alpha}\\le H$ for all $\\alpha$, whence $B_{*}\\le H$.  Thus $B_{*}=B$.\n\nTherefore the sequence stabilises below $\\kappa^{+}$ and its limit is $B$.\n\nStep 4.  Transfinite construction of the greatest fixed point $C$.\n\nDefine  \n\n\\[\nC_{0}=G,\\qquad\nC_{\\alpha+1}= \\bigcap_{\\sigma\\in\\Lambda^{<\\omega}}\\Phi_{\\sigma}(C_{\\alpha}),\\qquad\nC_{\\lambda}= \\bigcap_{\\beta<\\lambda}C_{\\beta}\\quad(\\lambda\\text{ limit}).\n\\]\n\n4a.  Descendingness.  \nBecause $C_{\\alpha}\\in\\text{Post}$ (next paragraph) and $\\Phi_{\\sigma}(C_{\\alpha})\\le C_{\\alpha}$, we have $C_{\\alpha+1}\\le C_{\\alpha}$.  Limits are meets, so the chain is descending.\n\n4b.  $C_{\\alpha}\\in\\text{Post}$ for all $\\alpha$ (transfinite induction).  \nBase $\\alpha=0$ is clear.  \nSuccessor: assume $C_{\\alpha}\\in\\text{Post}$.  For any $\\tau$\n\n\\[\n\\Phi_{\\tau}(C_{\\alpha+1})\n   =\\Phi_{\\tau}\\Bigl(\\bigcap_{\\sigma}\\Phi_{\\sigma}(C_{\\alpha})\\Bigr)\n   =\\bigcap_{\\sigma}\\Phi_{\\tau}\\bigl(\\Phi_{\\sigma}(C_{\\alpha})\\bigr)\n   =\\bigcap_{\\sigma}\\Phi_{\\sigma}\\bigl(\\Phi_{\\tau}(C_{\\alpha})\\bigr)\n   \\le\\bigcap_{\\sigma}\\Phi_{\\sigma}(C_{\\alpha})=C_{\\alpha+1},\n\\]\n\nwhere arbitrary-meet preservation and commutativity are used.  \nThus $C_{\\alpha+1}\\in\\text{Post}$.  \nLimit: meet-closure of $\\text{Post}$.\n\n4c.  Stabilisation below $\\kappa^{+}$.  \nA strictly descending chain of subgroups has length at most $\\kappa$, so $C_{\\beta}=C_{\\beta+1}$ for some $\\beta<\\kappa^{+}$.  Put $C_{*}:=C_{\\beta}$.\n\n4d.  $\\Phi$–stability of the limit.  \nBecause $C_{*}= \\bigcap_{\\sigma}\\Phi_{\\sigma}(C_{*})$, we have $C_{*}\\le\\Phi_{\\sigma}(C_{*})$; since $C_{*}\\in\\text{Post}$, also $\\Phi_{\\sigma}(C_{*})\\le C_{*}$.  Hence $\\Phi_{\\sigma}(C_{*})=C_{*}$.\n\n4e.  Maximality.  \nLet $H$ be $\\Phi$–stable.  Trivially $H\\le C_{0}$.  If $H\\le C_{\\alpha}$, then for every $\\sigma$\n\n\\[\nH=\\Phi_{\\sigma}(H)\\le\\Phi_{\\sigma}(C_{\\alpha}),\n\\]\n\nso $H\\le\\bigcap_{\\sigma}\\Phi_{\\sigma}(C_{\\alpha})=C_{\\alpha+1}$.  By transfinite induction $H\\le C_{\\alpha}$ for all $\\alpha$, hence $H\\le C_{*}$.  Therefore $C_{*}=C$.\n\nConsequently each sequence stabilises before stage $\\kappa^{+}$ and yields the corresponding extremal $\\Phi$–stable subgroup, completing part (c) and the proof.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.491373",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher-dimensional data.  \n   The original problem involves a single order-preserving map; the enhanced variant handles an arbitrarily large *family* of such maps, indexed by all finite words over an infinite alphabet Λ.\n\n2. Additional structure and interaction.  \n   The maps are required to commute and to satisfy consistency with respect to initial segments, forcing the solver to manage simultaneous constraints instead of a single one.\n\n3. Top–level lattice arguments plus transfinite methods.  \n   The solution needs the full machinery of complete lattices, transfinite induction, and cardinality bounds (|Sub(G)|) to control stabilisation, well beyond the finite, one-shot Knaster–Tarski argument.\n\n4. Multiple goals.  \n   Besides mere existence of a fixed point, the problem demands identification of *both* extremal fixed points, their explicit construction via transfinite iteration, and proof of maximality/minimality.\n\n5. Non-trivial continuity hypothesis.  \n   Condition (3) forces the solver to establish that unions of chains remain within the relevant sublattices; overlooking this breaks the Zorn and iteration arguments.\n\nAll these additions substantially raise the technical bar, require deeper knowledge (complete lattices, transfinite recursion, cardinal arithmetic), and preclude solving by a single short “take the union” trick that suffices in the original."
      }
    },
    "original_kernel_variant": {
      "question": "Let $G$ be an arbitrary (possibly infinite) group and write \n\n\\[\n\\operatorname{Sub}(G)=\\{\\,H\\le G\\,\\},\n\\]\n\nthe complete lattice of all subgroups of $G$, ordered by inclusion.  \nFix a non–empty index-set $\\Lambda$ and denote by $\\Lambda^{<\\omega}$ the set of all finite words  \n$\\sigma=(\\lambda_{1},\\dots,\\lambda_{n})$ (the empty word $\\varnothing$ is allowed).\n\nFor every $\\sigma\\in\\Lambda^{<\\omega}$ an order–preserving \\emph{complete} lattice endomorphism  \n\n\\[\n\\Phi_{\\sigma}\\colon\\operatorname{Sub}(G)\\longrightarrow\\operatorname{Sub}(G)\n\\]\n\nis given, i.e. $\\Phi_{\\sigma}$ preserves \\emph{all} joins and meets that exist in the lattice.  \nThese maps satisfy the structural requirement  \n\n\\[\n\\text{(C)\\quad Commutativity}\\qquad   \n\\Phi_{\\sigma}\\circ\\Phi_{\\tau}=\\Phi_{\\tau}\\circ\\Phi_{\\sigma}\\qquad\n\\text{for all finite words }\\sigma,\\tau.\n\\]\n\nA subgroup $H\\le G$ is called \\emph{$\\Phi$–stable} if $\\Phi_{\\sigma}(H)=H$ for every finite word $\\sigma$.\n\n(a) Prove that at least one $\\Phi$–stable subgroup of $G$ exists.\n\n(b) Put  \n\n\\[\n\\text{Pre}:=\\bigl\\{\\,H\\le G:\\; H\\le \\Phi_{\\sigma}(H)\\ \\text{for every } \\sigma \\bigr\\},\\qquad\n\\text{Post}:=\\bigl\\{\\,H\\le G:\\; \\Phi_{\\sigma}(H)\\le H\\ \\text{for every } \\sigma \\bigr\\},\n\\]\n\nand define  \n\n\\[\nB:=\\bigcap\\text{Post},\\qquad\nC:=\\left\\langle\\bigcup\\text{Pre}\\right\\rangle .\n\\]\n\nProve that $B$ is the least and $C$ the greatest $\\Phi$–stable subgroup of $G$.\n\n(c) Let $\\kappa:=|\\operatorname{Sub}(G)|$.  \nShow that $B$ and $C$ are obtained in fewer than $\\kappa^{+}$ transfinite steps by the monotone constructions  \n\nLeast fixed point  \n\n\\[\nB_{0}=\\{1\\},\\qquad\nB_{\\alpha+1}= \\bigvee_{\\sigma\\in\\Lambda^{<\\omega}}\\Phi_{\\sigma}(B_{\\alpha}),\\qquad\nB_{\\lambda}= \\bigvee_{\\beta<\\lambda}B_{\\beta}\\quad(\\lambda\\text{ limit});\n\\]\n\nGreatest fixed point  \n\n\\[\nC_{0}=G,\\qquad\nC_{\\alpha+1}= \\bigcap_{\\sigma\\in\\Lambda^{<\\omega}}\\Phi_{\\sigma}(C_{\\alpha}),\\qquad\nC_{\\lambda}= \\bigcap_{\\beta<\\lambda}C_{\\beta}\\quad(\\lambda\\text{ limit}).\n\\]\n\nProve that each transfinite sequence stabilises after fewer than $\\kappa^{+}$ steps and that its limit equals, respectively, $B$ and $C$.\n\n(The three parts together constitute a Knaster–Tarski fixed-point theorem for an arbitrary commuting family of complete lattice endomorphisms on $\\operatorname{Sub}(G)$.)\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Write $\\langle S\\rangle$ for the subgroup generated by a set $S$ and, for every family \n$\\mathcal H\\subseteq\\operatorname{Sub}(G)$, set  \n\n\\[\n\\bigvee\\mathcal H:=\\left\\langle\\bigcup_{H\\in\\mathcal H}H\\right\\rangle,\\qquad\n\\bigwedge\\mathcal H:=\\bigcap_{H\\in\\mathcal H}H .\n\\]\n\nStep 1.  Two auxiliary classes.  \n\n\\[\n\\text{Pre}:=\\{\\,H\\le G:H\\le\\Phi_{\\sigma}(H)\\ \\forall\\sigma\\,\\},\\qquad\n\\text{Post}:=\\{\\,H\\le G:\\Phi_{\\sigma}(H)\\le H\\ \\forall\\sigma\\,\\}.\n\\]\n\nBoth classes are non-empty: $\\{1\\}\\in\\text{Pre}$ and $G\\in\\text{Post}$.\n\n1a.  $\\text{Pre}$ is closed under arbitrary joins.  \nLet $\\mathcal H\\subseteq\\text{Pre}$ and put $K:=\\bigvee\\mathcal H=\\langle\\bigcup\\mathcal H\\rangle$.  \nFix $\\sigma$.  Each generator $g$ of $K$ lies in some $H\\in\\mathcal H$, hence  \n\n\\[\ng\\in H\\le\\Phi_{\\sigma}(H)\\subseteq\\Phi_{\\sigma}(K).\n\\]\n\nSince $\\Phi_{\\sigma}(K)$ is a subgroup, it contains every element of $K$, giving $K\\le\\Phi_{\\sigma}(K)$; thus $K\\in\\text{Pre}$.\n\n1b.  $\\text{Post}$ is closed under arbitrary meets.  \nLet $\\mathcal H\\subseteq\\text{Post}$ and set $L:=\\bigwedge\\mathcal H=\\bigcap\\mathcal H$.  \nFor each $\\sigma$,\n\n\\[\n\\Phi_{\\sigma}(L)=\\Phi_{\\sigma}\\!\\bigl(\\bigcap\\mathcal H\\bigr)\n               =\\bigcap_{H\\in\\mathcal H}\\Phi_{\\sigma}(H)\n               \\le\\bigcap_{H\\in\\mathcal H}H=L,\n\\]\n\nwhere the equality uses the fact that $\\Phi_{\\sigma}$ preserves arbitrary meets.  \nHence $L\\in\\text{Post}$.\n\nStep 2.  Extremal $\\Phi$–stable subgroups.  \nDefine  \n\n\\[\nB:=\\bigcap\\text{Post},\\qquad C:=\\bigvee\\text{Pre}.\n\\]\n\n2a.  $C$ is $\\Phi$–stable.  \nFix $\\sigma$.  Because every $H\\in\\text{Pre}$ satisfies $H\\le\\Phi_{\\sigma}(H)$, the join $C$ fulfils $C\\le\\Phi_{\\sigma}(C)$.  \nConversely, $\\Phi_{\\sigma}$ sends $\\text{Pre}$ into itself: if $H\\in\\text{Pre}$ and $\\tau$ is arbitrary then  \n\n\\[\n\\Phi_{\\tau}\\bigl(\\Phi_{\\sigma}(H)\\bigr)\n        =\\Phi_{\\sigma}\\bigl(\\Phi_{\\tau}(H)\\bigr)\n        \\ge\\Phi_{\\sigma}(H),\n\\]\n\nso $\\Phi_{\\sigma}(H)\\in\\text{Pre}$.  Join-closure of $\\text{Pre}$ yields $\\Phi_{\\sigma}(C)\\in\\text{Pre}$, whence $\\Phi_{\\sigma}(C)\\le C$.  Hence $\\Phi_{\\sigma}(C)=C$.\n\n2b.  $B$ is $\\Phi$–stable.  \nDually, $\\text{Post}$ is meet-closed and $\\Phi_{\\sigma}(\\text{Post})\\subseteq\\text{Post}$.  Therefore $\\Phi_{\\sigma}(B)\\in\\text{Post}$.  Because $B$ is the meet of $\\text{Post}$, we have  \n\n\\[\nB\\le\\Phi_{\\sigma}(B)\\le B,\n\\]\n\nso $\\Phi_{\\sigma}(B)=B$.\n\n2c.  Extremality.  Every $\\Phi$–stable subgroup lies in both $\\text{Pre}$ and $\\text{Post}$, hence $B\\le H\\le C$ for all $\\Phi$–stable $H$.\n\nParts (a) and (b) are complete.\n\nStep 3.  Transfinite construction of the least fixed point $B$.\n\nLet $\\kappa:=|\\operatorname{Sub}(G)|$ and define  \n\n\\[\nB_{0}=\\{1\\},\\qquad\nB_{\\alpha+1}= \\bigvee_{\\sigma\\in\\Lambda^{<\\omega}}\\Phi_{\\sigma}(B_{\\alpha}),\\qquad\nB_{\\lambda}= \\bigvee_{\\beta<\\lambda}B_{\\beta}\\quad(\\lambda\\text{ limit}).\n\\]\n\n3a.  Ascendingness.  \nBecause $B_{\\alpha}\\in\\text{Pre}$ (proved next) we have $B_{\\alpha}\\le\\Phi_{\\sigma}(B_{\\alpha})$ for every $\\sigma$, hence $B_{\\alpha}\\le B_{\\alpha+1}$.  Limits are joins, so the chain is ascending.\n\n3b.  $B_{\\alpha}\\in\\text{Pre}$ for all $\\alpha$ (transfinite induction).  \nBase $\\alpha=0$ is clear.  \nSuccessor: if $B_{\\alpha}\\in\\text{Pre}$ then each $\\Phi_{\\sigma}(B_{\\alpha})\\in\\text{Pre}$, and join-closure gives $B_{\\alpha+1}\\in\\text{Pre}$.  \nLimit: join-closure again.\n\n3c.  Stabilisation before stage $\\kappa^{+}$.  \nAn ascending chain of subgroups has length at most $\\kappa$, so some $\\alpha<\\kappa^{+}$ satisfies $B_{\\alpha}=B_{\\alpha+1}$.  Set $B_{*}:=B_{\\alpha}$.\n\n3d.  $\\Phi$–stability of the limit.  \nBy definition $B_{*}=\\bigvee_{\\sigma}\\Phi_{\\sigma}(B_{*})$, hence $\\Phi_{\\sigma}(B_{*})\\le B_{*}$ for every $\\sigma$.  Because $B_{*}\\in\\text{Pre}$ the reverse inclusion holds as well, yielding $\\Phi_{\\sigma}(B_{*})=B_{*}$.\n\n3e.  Minimality.  \nLet $H$ be $\\Phi$–stable.  By induction on $\\alpha$ and monotonicity of each $\\Phi_{\\sigma}$ we get $B_{\\alpha}\\le H$ for all $\\alpha$, whence $B_{*}\\le H$.  Thus $B_{*}=B$.\n\nTherefore the sequence stabilises below $\\kappa^{+}$ and its limit is $B$.\n\nStep 4.  Transfinite construction of the greatest fixed point $C$.\n\nDefine  \n\n\\[\nC_{0}=G,\\qquad\nC_{\\alpha+1}= \\bigcap_{\\sigma\\in\\Lambda^{<\\omega}}\\Phi_{\\sigma}(C_{\\alpha}),\\qquad\nC_{\\lambda}= \\bigcap_{\\beta<\\lambda}C_{\\beta}\\quad(\\lambda\\text{ limit}).\n\\]\n\n4a.  Descendingness.  \nBecause $C_{\\alpha}\\in\\text{Post}$ (next paragraph) and $\\Phi_{\\sigma}(C_{\\alpha})\\le C_{\\alpha}$, we have $C_{\\alpha+1}\\le C_{\\alpha}$.  Limits are meets, so the chain is descending.\n\n4b.  $C_{\\alpha}\\in\\text{Post}$ for all $\\alpha$ (transfinite induction).  \nBase $\\alpha=0$ is clear.  \nSuccessor: assume $C_{\\alpha}\\in\\text{Post}$.  For any $\\tau$\n\n\\[\n\\Phi_{\\tau}(C_{\\alpha+1})\n   =\\Phi_{\\tau}\\Bigl(\\bigcap_{\\sigma}\\Phi_{\\sigma}(C_{\\alpha})\\Bigr)\n   =\\bigcap_{\\sigma}\\Phi_{\\tau}\\bigl(\\Phi_{\\sigma}(C_{\\alpha})\\bigr)\n   =\\bigcap_{\\sigma}\\Phi_{\\sigma}\\bigl(\\Phi_{\\tau}(C_{\\alpha})\\bigr)\n   \\le\\bigcap_{\\sigma}\\Phi_{\\sigma}(C_{\\alpha})=C_{\\alpha+1},\n\\]\n\nwhere arbitrary-meet preservation and commutativity are used.  \nThus $C_{\\alpha+1}\\in\\text{Post}$.  \nLimit: meet-closure of $\\text{Post}$.\n\n4c.  Stabilisation below $\\kappa^{+}$.  \nA strictly descending chain of subgroups has length at most $\\kappa$, so $C_{\\beta}=C_{\\beta+1}$ for some $\\beta<\\kappa^{+}$.  Put $C_{*}:=C_{\\beta}$.\n\n4d.  $\\Phi$–stability of the limit.  \nBecause $C_{*}= \\bigcap_{\\sigma}\\Phi_{\\sigma}(C_{*})$, we have $C_{*}\\le\\Phi_{\\sigma}(C_{*})$; since $C_{*}\\in\\text{Post}$, also $\\Phi_{\\sigma}(C_{*})\\le C_{*}$.  Hence $\\Phi_{\\sigma}(C_{*})=C_{*}$.\n\n4e.  Maximality.  \nLet $H$ be $\\Phi$–stable.  Trivially $H\\le C_{0}$.  If $H\\le C_{\\alpha}$, then for every $\\sigma$\n\n\\[\nH=\\Phi_{\\sigma}(H)\\le\\Phi_{\\sigma}(C_{\\alpha}),\n\\]\n\nso $H\\le\\bigcap_{\\sigma}\\Phi_{\\sigma}(C_{\\alpha})=C_{\\alpha+1}$.  By transfinite induction $H\\le C_{\\alpha}$ for all $\\alpha$, hence $H\\le C_{*}$.  Therefore $C_{*}=C$.\n\nConsequently each sequence stabilises before stage $\\kappa^{+}$ and yields the corresponding extremal $\\Phi$–stable subgroup, completing part (c) and the proof.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.411161",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher-dimensional data.  \n   The original problem involves a single order-preserving map; the enhanced variant handles an arbitrarily large *family* of such maps, indexed by all finite words over an infinite alphabet Λ.\n\n2. Additional structure and interaction.  \n   The maps are required to commute and to satisfy consistency with respect to initial segments, forcing the solver to manage simultaneous constraints instead of a single one.\n\n3. Top–level lattice arguments plus transfinite methods.  \n   The solution needs the full machinery of complete lattices, transfinite induction, and cardinality bounds (|Sub(G)|) to control stabilisation, well beyond the finite, one-shot Knaster–Tarski argument.\n\n4. Multiple goals.  \n   Besides mere existence of a fixed point, the problem demands identification of *both* extremal fixed points, their explicit construction via transfinite iteration, and proof of maximality/minimality.\n\n5. Non-trivial continuity hypothesis.  \n   Condition (3) forces the solver to establish that unions of chains remain within the relevant sublattices; overlooking this breaks the Zorn and iteration arguments.\n\nAll these additions substantially raise the technical bar, require deeper knowledge (complete lattices, transfinite recursion, cardinal arithmetic), and preclude solving by a single short “take the union” trick that suffices in the original."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}