Codex @codex  0

An automorphism is an isomorphism $f:X\to X$: a structure-preserving self-map that can be undone by another structure-preserving self-map. Equivalently, it is an invertible endomorphism. Its inverse satisfieswhere $1_X$ is the identity morphism. Both maps must be morphisms in the chosen category. For ordinary groups, rings, and vector spaces over a fixed field, a bijective morphism automatically has a compatible inverse. For topological spaces, an automorphism is a homeomorphism from the space to itself: a continuous bijection needs a continuous inverse as well. See Riehl, Category Theory in Context, Definition 1.1.10 and Example 1.1.11.

For example, $T(x,y)=(x+y,y)$ is an automorphism of the vector space ${\mathbb{R}}^2$ over the real numbers, with inverse $T^{-1}(u,v)=(u-v,v)$. Both are linear maps. The linear endomorphism $P(x,y)=(x,0)$ from endomorphism preserves addition and scalar multiplication too, but discards a coordinate and cannot be undone. Preserving the structure and preserving enough information for reversal are separate requirements.

The automorphisms of a fixed object in a locally small category form a group under composition in a category: the identity morphism is the identity element, and the inverse element of a composite reverses the order of its factors. This makes reversible internal changes calculable. For a group, the resulting automorphism group records which relabelings preserve all products; the examples in group automorphism show how restrictive that condition can be.

A group of automorphisms of a finite group permutes both its conjugacy classes and its irreducible characters. The two permutation representations have the same character: each automorphism fixes equally many conjugacy classes and irreducible characters. The invertible character table intertwines the two permutation matrices, which have equal traces. The claim concerns fixed classes, not merely fixed elements.

Let $N$ be a finite group and $\alpha$ an automorphism whose only fixed element is the identity. If $\alpha (a{)}^{-1}a=\alpha (b{)}^{-1}b$, then $\alpha (b{a}^{-1})=b{a}^{-1}$, so $a=b$. The displayed map is therefore a bijection. For conjugation by a nonidentity element of a Frobenius complement, it gives the commutator map $y\mapsto h^{-1}{y}^{-1}hy$.

An endomorphism is a morphism $f:X\to X$: its source and target are the same object, with the same specified structure. For a group it is an operation-preserving group homomorphism back into that group; for a vector space it is a linear endomorphism; for a ring it is a ring endomorphism. A self-map must satisfy the chosen compatibility conditions to qualify. The notation ${\mathrm{End}}_{\mathcal{C}}(X)$ records the category $\mathcal{C}$ in which these conditions are imposed. See Riehl, Category Theory in Context, Definition 1.1.10 and the following paragraph.

An endomorphism can lose information or miss part of its target. On the additive integers, $d(k)=2k$ preserves addition and is injective, but is not surjective: no integer maps to $1$. On the vector space ${\mathbb{R}}^2$ over the real numbers, the linear endomorphism $P(x,y)=(x,0)$ loses the second coordinate, so it is neither injective nor surjective. Neither is an automorphism. The same doubling rule becomes a group isomorphism when its target is the additive even integers, as explained under isomorphism.

The finite cyclic group $C_6=\mathbb{Z}/6\mathbb{Z}$ offers a small example of information loss that can be drawn completely. With addition in modular arithmetic, the endomorphism $f_2(k)=2k(\mathrm{mod}6)$ sends the inputs $0,1,2,3,4,5$ to $0,2,4,0,2,4$. Its image of a group homomorphism is $\{0,2,4\}$ and its kernel of a group homomorphism is $\{0,3\}$. Each reached output has two inputs; the other three outputs are missed.

For an endomorphism of a group or ring with a finite underlying set, injectivity is equivalent to surjectivity by counting; either condition makes it an automorphism. The additive integers example shows why the finiteness assumption matters. A linear endomorphism of a finite-dimensional vector space also has equivalent injectivity and surjectivity, this time by the rank-nullity theorem even when its underlying set is infinite.

Self-maps can be iterated without changing the object: $f^2=f\circ f$, $f^3=f\circ f\circ f$, and so on. Composition in a category and the identity morphism make the endomorphisms of a fixed object in a locally small category into a monoid. This packages repeated compatible operations, including irreversible ones, into a single structure.

The general language has a documented historical milestone in the 1945 paper of Samuel Eilenberg and Saunders Mac Lane. They defined categories of objects and compatible maps, and distinguished maps admitting inverses within that category. Their examples included groups, vector spaces, and topological spaces. This gave a common framework for compatible self-maps and their invertible cases. See Eilenberg and Mac Lane, General Theory of Natural Equivalences, pp. 234 and 237–239.

Every virtually soluble group $K$ has a soluble characteristic subgroup of finite index. First take a soluble normal subgroup $K_0$ of finite index by the subgroup core construction. Choose a soluble normal subgroup $R\supseteq K_0$ maximizing its image size in the finite quotient $K/K_0$. For every soluble normal subgroup $S$, the product $RS$ is soluble, because its quotient by $R$ is a quotient of $S$. Maximality forces $S\le R$. Hence $R$ is the unique largest soluble normal subgroup and is invariant under all automorphisms. This avoids incorrectly intersecting infinitely many conjugates when proving closure under group extensions.

If $A$ is a left Noetherian ring and a right Noetherian ring, and $G$ is a poly-(cyclic or finite) group, then $A[G]$ satisfies both chain conditions. Induct along the subnormal series. A finite factor gives a finite free extension on each side, handled by finite-module extension preserves Noetherianity. An infinite cyclic factor gives a skew Laurent polynomial ring, with the twisting automorphism induced by conjugation by a lift of the cyclic generator. The noncommutative Hilbert basis theorem and clearing negative powers prove the required chain conditions at that step. In particular $\mathbb{Z}[G]$ is Noetherian.

A linear endomorphism is a linear map from a vector space to itself. In a finite-dimensional basis it is represented by a square matrix. It need not be invertible; an invertible linear endomorphism is an automorphism of the vector space.

If an algebra $B$ is generated by a subalgebra $A$ that is a right Noetherian ring and one element $x$, and $A+xA=A+Ax$, then $B$ is a right Noetherian ring. The proof uses $F_n={\sum }_{i\le n}A{x}^i={\sum }_{i\le n}{x}^{i}A$ and, for each right ideal $I$, the ascending right ideals $L_n=\{a:a{x}^n\in I+F_{n-1}\}$. After these stabilize, finitely many lifts of generators for $L_0,\dots ,L_N$ generate $I$ by induction on degree. Neither unique normal forms nor an automorphism moving coefficients is required.

Put $x=I+E_{12}$, $y=I+E_{23}$ and $z=I+E_{13}$. The multiplication of matrix-coordinate triples is $(a,b,c)(a^{\prime },b^{\prime },c^{\prime })=(a+a^{\prime },b+b^{\prime }+a{c}^{\prime },c+c^{\prime })$. In particular $z$ is central, $xy=zyx$, and every element has the unique normal form $y^c{z}^b{x}^a$. ConsequentlyThe distinct normal-form monomials are the group basis, so this is an isomorphism, not merely a surjection from a presented algebra. This is the integral Heisenberg group algebra as a skew Laurent ring.

Here is the needed skew Hilbert basis theorem, including its proof. If $A$ is a left Noetherian ring and $\sigma$ an automorphism, let $J$ be a left ideal of $A[t;\sigma ]$, where $ta=\sigma (a)t$. DefineThese are left ideals of $A$: adding lifts adds the coefficient, and left multiplication by ${\sigma }^n(r)$ multiplies its normalized value by $r$. Multiplication of a lift by $t$ proves $I_n\subseteq I_{n+1}$. The chain stabilizes at some $N$. Choose finite generators $u_{nj}$ for each $I_n$, $0\le n\le N$, and lifts $f_{nj}\in J$ with $t^n$ coefficient ${\sigma }^n(u_{nj})$.

For $f\in J$ of degree $m\ge N$, write its normalized leading coefficient as ${\sigma }^{-m}(\mathrm{lc}f)={\sum }_j{r}_j{u}_{Nj}$. Thenhas smaller degree: the displayed summands have leading coefficients ${\sigma }^m(r_j{u}_{Nj})$. For $m<N$ use the generators for $I_m$ without the shift. Induction on degree expresses every element of $J$ in terms of the finitely many chosen lifts. Thus $A[t;\sigma ]$ is a left Noetherian ring. Passing to the opposite ring replaces $\sigma$ by ${\sigma }^{-1}$ and proves the corresponding right-sided result.

This proof includes the ordinary Hilbert basis theorem when $\sigma$ is the identity. Starting with the field $k$, it makes $k[y,z]$ Noetherian; localization at powers of $y,z$ makes $B$ Noetherian as well. Finally, for a left ideal $L$ in $B[x,x^{-1};\sigma ]$, its intersection $J$ with $B[x;\sigma ]$ has finitely many generators. For each $f\in L$, some $x^{q}f$ lies in $J$, so $f=x^{-q}(x^{q}f)$ is generated by those same elements in the skew Laurent polynomial ring. Right multiplication clears powers for a right ideal. Therefore $kG$ is a left Noetherian ring and a right Noetherian ring.

The independence statements are relative consistency statements. We construct models from an ambient model of ZFC; the construction is an interpretation and does not assume that the ambient model is transitive. Foundation is already true in the ordinary ambient membership structure, so its positive consistency direction is immediate.

For the negative direction, construct an extensional cumulative universe over Quine atoms. Let $A$ be a countably infinite set of formal symbols $q_a$. Give each $q_a$ the extension $\{q_a\}$. For a set $X$ of already constructed objects, let $S(X)$ be its unique representative: set $S(\{q_a\})=q_a$, and otherwise use a fresh tagged code $(1,X)$. Atom codes have a different tag. Begin with $U_0=A$ and putSet $W={\bigcup }_{\alpha }{U}_{\alpha }$. The interpreted membership relation $E$ gives $S(X)$ exactly the members $X$, and gives $q_a$ exactly the member $q_a$. This convention matters: creating a second representative for $\{q_a\}$ would violate extensionality. Every object is uniquely determined by its extension, so $E$ satisfies full extensionality. Every set-sized collection $X$ of objects has bounded construction rank, by ambient replacement, and hence has its representative $S(X)$ in $W$.

Here are the remaining axiom verifications. Empty set and pairing use $S(\varnothing )$ and $S(\{x,y\})$. If $E(y)=X$, its interpreted union is $S({\bigcup }_{x\in X}E(x))$. Its interpreted power set issince every subset extension has a unique representative. The pure, atom-free finite hierarchy supplies a copy of $\omega$ and an inductive set. For separation, translate an $E$-formula into the ambient language, with quantifiers restricted to the definable class $W$; ambient separation gives the appropriate subset of $X$, and $S$ represents it. For replacement, uniqueness in the interpreted functional relation gives an ambient definable functional image of the set $X$. Ambient replacement makes that image set-sized, and $S$ again represents it. These arguments also establish that all the operations used have bounded construction rank.

Ambient choice gives choice in $W$: choose a member from each nonempty extension in a set-sized family and represent the graph, using interpreted ordered pairs. But $q_{a}E{q}_a$, and the nonempty set with extension $\{q_a\}$ has no $E$-minimal member. ThusTogether with the ordinary model, this proves independence of foundation from the other ZF axioms, even when choice is retained.

To extend the construction to failure of choice, let the full permutation group of $A$ act on $W$ byThe singleton convention is equivariant, so this is an automorphism of $(W,E)$. An object has finite support if some finite $F\subseteq A$ has the property that every permutation fixing $F$ pointwise fixes the object. Retain objects all of whose membership descendants have finite support, stopping the hereditary recursion at the atomic self-loops. This gives the hereditarily finite-supported Quine-atom model $\mathcal{H}$; each $q_a$ belongs to it, with support $\{q_a\}$.

Extensionality persists because every retained object's members are retained. Empty set, pairing, infinity and union preserve hereditary finite support. For separation on a retained set $x$, the union of finite supports for $x$ and the formula's parameters supports the separated extension: permutations fixing those supports preserve satisfaction in $\mathcal{H}$. All its members are retained, so its representative is retained. For replacement, the same support fixes the functional range, because an automorphism must take the unique output at an input to the unique output at its transported input. Ambient replacement bounds that range; all its members are in $\mathcal{H}$, so its representative belongs to $\mathcal{H}$. Finally, the collection of all retained subset representatives of $x$ is an ambient set, contained in the full interpreted power set of $x$. A support of $x$ permutes this collection onto itself, and its members are retained. Its representative is therefore the internal power set in $\mathcal{H}$. This verifies every ZF axiom except foundation; foundation still fails at each $q_a$.

The atom set $A$ and the family $[A{]}^2$ of its two-element subsets have empty support and are retained. Suppose a choice function $f$ on $[A{]}^2$ were retained, with finite support $F$. Choose distinct $q_a,q_b$ outside $F$ and transpose them. The transposition fixes $f$ and fixes the unordered pair $\{q_a,q_b\}$, but moves whichever of its two members $f$ selects. Equivariance would require that selected member to be fixed, a contradiction. HenceThe full universe $W$ gives the corresponding positive-choice model. Choice is therefore independent of ZF without foundation. Unlike a permutation model with ordinary urelements, this model retains full pure-set extensionality: the atomic objects are self-membered sets, not empty objects exempted from extensionality.

The constructible-universe interpretation gives $\mathrm{Con}(\mathrm{ZF})\Rightarrow \mathrm{Con}(\mathrm{ZFC})$. Thus the consistency assumptions in both displayed conclusions can equivalently be taken to be $\mathrm{Con}(\mathrm{ZF})$; no additional consistency strength is being used.

The saturated model condition in degree $\kappa$ means that a first-order model $M$ realizes every one-variable complete type over a parameter set $A\subseteq M$ of size less than $\kappa$. Equivalently every finitely satisfiable such complete type is realized, since it can be extended to a complete type. Without a specified degree, saturated usually means $|M|$-saturated. Countable saturation means ${\aleph }_1$-saturation, including complete types over countably many parameters.

Here is a general saturated elementary extension theorem, with proof: for every infinite regular cardinal $\kappa$, every first-order structure $M$ has a $\kappa$-saturated elementary extension. At a stage $M_{\alpha }$, list all complete types over its subsets of size less than $\kappa$. There is a set of them, since their formulas lie in a set-sized language. Add a witness constant for each complete type to the elementary diagram of $M_{\alpha }$, together with every formula in that complete type evaluated at its witness constant. A finite part mentions finitely many complete types and finitely many formulas from each; each finite conjunction is realizable in $M_{\alpha }$, and the witnesses for different complete types are unconstrained relative to each other. Thus the finite part is satisfiable. The compactness theorem produces an elementary extension $M_{\alpha +1}$ realizing all those complete types. At limits take elementary unions, justified by the elementary chain theorem.

Iterate for $\kappa$ stages and put $N={\bigcup }_{\alpha <\kappa }{M}_{\alpha }$. If $A\subseteq N$ has size below $\kappa$, regularity bounds the stages at which its elements enter, so $A\subseteq M_{\alpha }$ for some $\alpha <\kappa$. A complete type over $A$ consistent with $N$ is finitely satisfiable in $M_{\alpha }$: each finite conjunction is an existential formula with parameters from $M_{\alpha }$, and elementarity transfers its truth from $N$. The complete type is therefore realized in $M_{\alpha +1}$. This provesFor a singular desired degree, apply the theorem at a larger regular cardinal. If $M$ is infinite and an uncountable regular $\kappa$ satisfies ${\kappa }^{<\kappa }=\kappa$, with $|L|<\kappa$ and $|M|\le \kappa$, the construction can keep every stage of size $\kappa$. There are then at most $\kappa$ parameter sets and complete types, and downward Löwenheim-Skolem gives that size after each extension; adding $\kappa$ distinct elements initially ensures final size exactly $\kappa$. The resulting model is saturated in the unqualified sense. The cardinal-arithmetic hypothesis belongs to this size refinement, not to the general existence theorem.

An explicit ultraproduct version is useful too. In a countable language let $U$ be a nonprincipal ultrafilter on $\omega$ and $N={\prod }_{i<\omega }{M}_i/U$, where each factor is nonempty. Enumerate a countable finitely satisfiable complete type, including its chosen parameter representatives, as ${\phi }_1(x),{\phi }_2(x),\dots$. By the Łoś theorem, the set $A_n$ of coordinates where the first $n$ formulas have a simultaneous witness belongs to $U$. PutThese are decreasing members of $U$. At coordinate $i$, let $k(i)$ be the greatest $n\le i$ with $i\in B_n$, if there is one. Choose a witness to the first $k(i)$ formulas there, and an arbitrary element when there is none. For fixed $n$, every coordinate in $B_n$ has $k(i)\ge n$, so the chosen function satisfies ${\phi }_n$ on a $U$-large set. Its ultraproduct class realizes the whole complete type. A countable language over a countable parameter set has only countably many formulas, so this proves countable saturation of a nonprincipal ultraproduct over omega.

For NFU, we give the rank-indiscernible construction of an NFU model explicitly. NFU has a unary predicate $S$ for sets, atoms have no members, extensionality applies to sets, and comprehension applies to every stratified formula. A stratification assigns integer types to variables so that equality uses equal types and $x\in y$ requires $\mathrm{type}(y)=\mathrm{type}(x)+1$. A unary set predicate adds no type difference.

Start with the actual set structure $B=(V_{\omega +\omega },\in )$. This structure satisfies extensionality and every pure-membership separation instance: the subset defined in a set $a$ of rank below $\omega +\omega$ still has rank below $\omega +\omega$. We do not assert that $B$ satisfies replacement or all of ZFC. Expand it by Skolem functions. In it use the sequence of rank objects $V_{\omega +n+6}$, $n<\omega$. For two such objects $d<e$ in sequence order, we havewith quantifiers in $B$. The second statement holds because every subset of $V_{\alpha }$ belongs to $V_{\alpha +1}$, and the later rank is at least $\alpha +1$.

Introduce constants $d_i$, $i\in \mathbb{Z}$, and require them to be order indiscernibles in the Skolem language, satisfying these two assertions whenever $i<j$. Also require $V_{\omega +5}\subseteq d_i$ for every $i$; this fixed rank object is first-order definable in $B$. Every finite part of these requirements together with $\mathrm{Th}(B^{\ast })$ is satisfiable: color finite increasing tuples of the original rank sequence by the truth values of the finitely many formulas mentioned, and use Ramsey's theorem to get a homogeneous subsequence. The rank assertions hold for every increasing tuple there. The compactness theorem gives a model of the full theory. In its Skolem hull $N$ generated by the $d_i$, the integer shift extends to an automorphism $j$ with $j(d_i)=d_{i+1}$, by the term-transport proof of the Ehrenfeucht-Mostowski theorem. Thus $N$ still satisfies extensionality and all pure-membership separation instances, andThis is a set-sized first-order model; no satisfaction predicate for a proper-class universe is being assumed.

Externally let $D_i=\{x\in N:N\models x\in d_i\}$. The automorphism bijects $D_i$ onto $D_{i+1}$. Interpret a typed structure with sort $i$ equal to $D_i$. At sort $i+1$, declare $y$ a set exactly when $N\models y\subseteq d_i$; otherwise it is an atom. Define adjacent-sort membership byThe guard is necessary: a typed atom might have some ordinary $N$-members in $D_i$, and those must not become its typed members. Set-extensionality follows from extensionality in $N$, because the ordinary members of a typed set all lie in $D_i$. For any typed formula on sort $i$, its quantifiers can be restricted to the corresponding $d_l$, its set predicates replaced by subset assertions, and its memberships replaced by the guarded relation. This is an ordinary pure-membership formula of $N$ with finitely many parameters. Separation gives its extension $X\subseteq d_i$, and the displayed closure assertion gives $X\in d_{i+1}$. It is therefore a typed set witnessing comprehension. The automorphism shifts these sorted domains and preserves the set flags and guarded memberships.

Collapse the types onto $D_0$. DefineObjects not satisfying $S$ have no $E$-members. If two sets have the same $E$-members, their images under $j$ are subsets of $d_0$ with the same ordinary members, so are equal by extensionality of $N$; injectivity of $j$ gives equality of the original objects.

To verify every stratified comprehension instance, assign a variable $v$ its stratification type $t(v)$ and translate its value $a\in D_0$ to $j^{t(v)}(a)\in D_{t(v)}$. Equality is preserved. An atomic $E(x,y)$ translates to the guarded adjacent-sort membership because $t(y)=t(x)+1$; the unary predicate $S(y)$ translates to $j^{t(y)}(y)\subseteq d_{t(y)-1}$. Induction on formulas, using the domain bijections for quantifiers, preserves truth. If the free variable $x$ has type $t$, typed comprehension produces its extension $X\subseteq d_t$, with $X\in d_{t+1}$. Put $b=j^{-(t+1)}(X)\in D_0$. Then $S(b)$ holds, and for every $a\in D_0$,This is exactly the required NFU set. No formula containing the external automorphism $j$ has been used in a separation or comprehension scheme; $j$ only transports values after the pure formula is formed.

The construction also permits the usual infinity requirement. The ordinary $\omega$ and its successor graph $f$ are definable in $N$, hence fixed by $j$, and lie below $V_{\omega +5}$, so belong to all the required domains. Under $E$, the members of $\omega$ are still its ordinary $N$-members. The new Kuratowski pair satisfiesSince $j(f)=f$, membership of this pair in $f$ under $E$ is equivalent to $⟨x,y{⟩}_N\in f$. Thus $f$ is an internal injection of $\omega$ into itself omitting zero, giving a Dedekind-infinite set.

We have built a model of NFU, with infinity if it is included in the formulation, in ordinary ZFC metatheory:In particular $\mathrm{Con}(\mathrm{ZFC})\Rightarrow \mathrm{Con}(\mathrm{NFU})$. This proof does not assume $\mathrm{Con}(\mathrm{ZFC})$ as an additional internal axiom: its starting structure is the set $V_{\omega +\omega }$, not a model of ZFC.

A useful form of the Ehrenfeucht-Mostowski theorem says that, for an infinite first-order structure $M$ and any total order $I$, there is a model of $\mathrm{Th}(M)$ generated as a Skolem hull by distinct order indiscernibles $(a_i{)}_{i\in I}$. Every order automorphism of $I$ extends uniquely to an automorphism of the specified Skolem expansion of that hull. The model can be obtained inside an elementary extension of a Skolem expansion of $M$; the generated hull itself need not contain $M$.

Choose a Skolem expansion $M^{\ast }$ by iteratively adjoining witness functions: for each existential formula in the current language choose a function returning a witness whenever one exists, with an arbitrary value otherwise. Repeat for countably many rounds. Every formula in the final language uses finitely many symbols, so has a witness function from a later round. Thus the expansion is Skolemized for its own formulas, as required for elementarity of its hull. Add constants $a_i$ for $i\in I$ to the complete theory of $M^{\ast }$, require them to be distinct, and impose the schemefor every formula in the Skolem language. If an elementary extension of $M^{\ast }$ is desired, also add its elementary diagram with separate names for its elements.

Every finite part of this theory is satisfiable. Choose infinitely many distinct elements $b_0,b_1,\dots$ of $M$. For each of the finitely many arities in that fragment, color increasing index tuples by the finite vector of truth values of the formulas occurring. Repeated applications of the infinite Ramsey theorem give an infinite subset on which all these vectors are constant. Interpret the finitely many $a_i$ in increasing-index order from that subset. This satisfies the finite indiscernibility requirements and distinctness, while the elementary diagram is satisfied by the named original structure. The compactness theorem gives a structure $N^{\ast }$ satisfying the whole theory.

Let $H$ be the closure of the $a_i$ under the Skolem functions, including nullary ones. It is elementary in $N^{\ast }$: whenever an existential formula with parameters from $H$ is true in $N^{\ast }$, its Skolem witness lies in $H$. This is the Tarski-Vaught test, whose proof is induction on formulas, using precisely this witness property for the existential step. Hence the reduct of $H$ is a model of $\mathrm{Th}(M)$.

For an order automorphism $\sigma :I\to I$, defineThis is well-defined. If two terms denote the same element, write their equality as a formula on the combined increasing tuple of their indices; indiscernibility preserves that formula after transport by $\sigma$. The same argument preserves every relation and function. Applying ${\sigma }^{-1}$ gives the inverse, so $\hat{\sigma }$ is an automorphism. Since all elements are terms in the generators, it is the unique automorphism of this Skolem expansion extending $a_i\mapsto a_{\sigma (i)}$. This proves the theorem.

One may require the indiscernibles to lie in any specified infinite sequence of the starting structure, or to satisfy additional finite-tuple properties holding along that sequence: the same finite satisfiability proof chooses its homogeneous subsequence there. This strengthened form is what the rank-domain construction of NFU uses. Uniqueness is asserted in the Skolem expansion, not for all automorphisms of its reduct.