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WEAK HEIRS, COHEIRS, AND THE ELLIS SEMIGROUPS

Published online by Cambridge University Press:  07 September 2023

ADAM MALINOWSKI
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL.GRUNWALDZKI 2, 50-384 WROCŁAW, POLAND E-mail: aadam.malinowski@gmail.com
LUDOMIR NEWELSKI*
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL.GRUNWALDZKI 2, 50-384 WROCŁAW, POLAND E-mail: aadam.malinowski@gmail.com
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Abstract

Assume $G\prec H$ are groups and ${\cal A}\subseteq {\cal P}(G),\ {\cal B}\subseteq {\cal P}(H)$ are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of the G-flow $S({\cal A})$ and the H-flow $S({\cal B})$. We apply these results in the model theoretic context. Namely, assume G is a group definable in a model M and $M\prec ^* N$. Using weak heirs and weak coheirs we point out some algebraic connections between the Ellis semigroups $S_{ext,G}(M)$ and $S_{ext,G}(N)$. Assuming every minimal left ideal in $S_{ext,G}(N)$ is a group we prove that the Ellis groups of $S_{ext,G}(M)$ are isomorphic to closed subgroups of the Ellis groups of $S_{ext,G}(N)$.

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© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

1 Introduction

Assume M is a model for a relational language L, $T=Th(M)$ and $G=G(M)$ is an infinite group $0$ -definable in M. All models of T we consider are elementary submodels of a monster model ${\cal M}$ of T. In this paper we are interested in topological dynamics of some G-flows of types over M. We assume the reader is familiar with the basic notions of topological dynamics [Reference Ellis1, 2, Reference Glasner4] and model theory [Reference Hodges5, Reference Lascar7, Reference Pillay13] as well as topological dynamics of definable groups [Reference Newelski9], although we shall recall some of them.

Let ${\mathrm {Def}}_{ext,G}(M)$ be the algebra of externally definable subsets of G and let $S_{ext,G}(M)$ be the Stone space of ultrafilters in ${\mathrm {Def}}_{ext,G}(M)$ , called external G-types over M. $S_{ext,G}(M)$ is a G-flow, in model theory it is a counterpart of the universal G-flow $\beta G$ in topological dynamics. In particular, the G-flow $S_{ext,G}(M)$ is isomorphic to its Ellis semigroup, hence it carries the induced semigroup operation $*$ that is continuous in the first coordinate (a semigroup like that is called left-continuous).

There is a question, how are the flows $S_{ext,G}(M)$ related for various models M of T, in particular, are the semigroups $(S_{ext,G}(M),*)$ and their Ellis groups related algebraically? [Reference Newelski10, Reference Newelski11] contain some results on these questions. The results of [Reference Newelski9] and [Reference Krupinski and Pillay6, Proposition 3.1] indicate that the quotient groups of $G({\cal M})$ by its model theoretic M-connected components are homomorphic images of the semigroup $S_{ext,G}(M)$ and also of its Ellis groups.

In order to compare the G-flow $S_{ext,G}(M)$ and the $G(N)$ -flow $S_{ext,G}(N)$ for $M,N\models T$ we need to be able to compare the algebras ${\mathrm {Def}}_{ext,G}(M)$ and ${\mathrm {Def}}_{ext,G}(N)$ . So it is natural to assume that $M\prec N$ , but it is not enough. We need a stronger relation between M and N, called $*$ -elementary inclusion $\prec ^*$ , we recall it later.

Assume $M\prec ^* N$ . In [Reference Newelski10, Theorem 4.1] we proved that if there is a generic type in $S_{ext,G}(N)$ (equivalently, $S_{ext,G}(N)$ has a unique minimal subflow [Reference Newelski9]), then the Ellis groups of $S_{ext,G}(M)$ are homomorphic images of subgroups of the Ellis groups of $S_{ext,G}(N)$ . Here we prove that if every minimal left ideal in $S_{ext,G}(N)$ is a group, then the Ellis groups of $S_{ext,G}(M)$ are isomorphic to closed subgroups of the Ellis groups of $S_{ext,G}(N)$ . The assumption that every minimal left ideal in $S_{ext,G}(N)$ is a group is dual to the assumption that $S_{ext,G}(N)$ contains a generic type.

We present our results in a general combinatorial set up, making the presentation easier. So we consider infinite groups $G\prec H$ , a G-algebra ${\cal A}\subseteq {\cal P}({\cal G})$ and an H-algebra ${\cal B}\subseteq {\cal P}(H)$ and under some additional assumptions we find some algebraic connections between the Ellis [semi]groups of $S({\cal A})$ and $S({\cal B})$ . The important tools in our proofs are the notions of weak heir from [Reference Newelski10] and weak coheir (introduced here). We think they are of independent interest, so we elaborate on them. We characterize them in the case where T is stable, by means of local forking. We provide some examples.

2 Preliminaries

Assume G is an infinite (discrete) group. We use e to denote the identity element of G and also of any other group. Assume that ${\cal A}$ is a G-algebra, that is an algebra of subsets of G closed under left translation by elements of G. In this case its Stone space $S({\cal A})$ is naturally a G-flow, with G acting on $S({\cal A})$ by left translation. For $A\in {\cal A}$ , $S({\cal A})\cap [A]$ denotes the set $\{p\in S({\cal A}):A\in p\}$ , a clopen subset of $S({\cal A})$ . When $p\in S({\cal A})$ then $cl(Gp)$ is the subflow of $S({\cal A})$ generated by p.

For $g\in G$ let $l_g:S({\cal A})\to S({\cal A})$ be the left translation by g. Let $E(S({\cal A}))$ be the topological closure of the set $\{l_g:g\in G\}$ in the space of functions $S({\cal A})\to S({\cal A})$ with the topology of pointwise convergence. $E(S({\cal A}))$ is a G-flow, where $g\cdot f = l_g\circ f$ for $g\in G$ and $f\in E(S({\cal A}))$ . $E(S({\cal A}))$ is also a compact left-continuous semigroup, with respect to composition of functions, called the Ellis semigroup of $S({\cal A})$ .

Assume S is a semigroup. We write $I\triangleleft S$ [ $I\triangleleft _m S$ ] when I is a [minimal] left ideal in S. We say that $u\in S$ is an idempotent if $u^2=u$ . The next fact is a fundamental structural result of Ellis.

Fact 2.1 [Reference Ellis2],[Reference Ellis, Ellis and Nerurkar3, Proposition 4.2].

Assume S is a compact Hausdorff left-continuous semigroup and let J be the set of idempotents in S.

$(1)$ Given $I\triangleleft _m S$ , the set $J(I)=I\cap J$ is non-empty.

$(2)$ For every $I\triangleleft _m S$ and $u\in J(I)$ , the set $uI$ is a subgroup of I and I is a disjoint union of such subgroups.

$(3)$ The groups $uI,I\triangleleft _m S,u\in J(I)$ , are all isomorphic.

$(4)$ If $a\in S$ and $I\triangleleft _m S$ , then $Ia\triangleleft _m S$ .

The groups $uI$ appearing in Fact 2.1 are called the Ellis subgroups of S. $E(S({\cal A}))$ is a compact left-continuous semigroup, hence Fact 2.1 applies. We call its Ellis subgroups the Ellis groups of the flow $S({\cal A})$ .

When ${\cal A}={\cal P}(G)$ , then $S({\cal A})=\beta G$ is naturally isomorphic to $E(\beta G)$ . This fact has been generalized in [Reference Newelski12] to some other G-algebras as follows.

Definition 2.2 [Reference Newelski12, Definition 2.2].

$(1)$ For $p\in S({\cal A})$ we define a function $d_p:{\cal A}\to {\cal P}(G)$ by

$$ \begin{align*}d_p(A)=\{g\in G: g^{-1}A\in p\}.\end{align*} $$

Clearly $d_p:{\cal A}\to {\cal P}(G)$ is a homomorphism of G-algebras.

$(2)$ We say that ${\cal A}$ is d-closed if ${\cal A}$ is closed under $d_p$ for every $p\in S({\cal A})$ , that is $d_p[{\cal A}]\subseteq {\cal A}$ . Notice that in this case $d_p$ belongs to ${\mathrm {End}}({\cal A})$ , the semigroup of endomorphisms of G-algebra ${\cal A}$ .

$(3)$ If ${\cal A}$ is d-closed, then let $d:S({\cal A})\to {\mathrm {End}}({\cal A})$ be the function mapping p to $d_p$ .

So the algebra ${\cal P}(G)$ is d-closed. By [Reference Newelski10, Lemma 1.2] also the G-algebra ${\mathrm {Def}}_{ext,G}(M)$ is d-closed. The next fact lists the basic properties of d-closed G-algebras.Footnote 1

Fact 2.3 [Reference Newelski12, Proposition 2.4].

Assume ${\cal A}$ is d-closed.

$(1)$ The function $d:S({\cal A})\to {\mathrm {End}}({\cal A})$ is a bijection.

$(2)$ d induces on $S({\cal A})$ a semigroup operation $*$ so that $d:S({\cal A})\to {\mathrm {End}}({\cal A})$ becomes an isomorphism of semigroups and for $p,q\in S({\cal A})$ we have $d_{p*q}=d_p\circ d_q$ . Also for $A\in {\cal A}$ we have

$$ \begin{align*}A\in p*q\iff d_q(A)\in p.\end{align*} $$

(3) For $p\in S({\cal A})$ let $l_p:S({\cal A})\to S({\cal A})$ be the left translation by p, that is, $l_p(q)=p*q$ . Then $E(S({\cal A}))=\{l_p:p\in S({\cal A})\}$ and the function $l:S({\cal A})\to E(S({\cal A}))$ mapping p to $l_p$ is an isomorphism of G-flows and of semigroups. In particular, $(S({\cal A}),*)$ is a compact left-continuous semigroup.

When ${\cal A}$ is d-closed and $p\in S({\cal A})$ , we associate with $d_p\in {\mathrm {End}}({\cal A})$ its kernel ${\mathrm {Ker}} d_p$ and image ${\mathrm {Im}} d_p$ . Clearly ${\mathrm {Ker}} d_p$ is a G-ideal in ${\cal A}$ and ${\mathrm {Im}} d_p$ is a G-subalgebra of ${\cal A}$ .

Fact 2.4 [Reference Newelski10, Reference Newelski12].

Assume ${\cal A}$ is d-closed.

$(1)$ Assume $p,q\in S({\cal A})$ . Then ${\mathrm {Ker}} d_p\subseteq {\mathrm {Ker}} d_q$ iff $cl(Gp)\supseteq cl(Gq)$ .

$(2)$ If S is a subgroup of $S({\cal A})$ , then the endomorphisms $d_p,p\in S$ have common kernel $K(S)={\mathrm {Ker}} d_p$ and image $R(S)={\mathrm {Im}} d_p$ . $R(S)$ is a section of the family of $K(S)$ -cosets in ${\cal A}$ , hence $R(S)\cong {\cal A} / K(S)$ . Also, every $d_p,p\in S$ restricted to $R(S)$ is a G-algebra automorphism of $R(S)$ and the mapping $p\mapsto d_p|_{R(S)}$ is a group monomorphism $S\to {\mathrm {Aut}}(R(S))\cong {\mathrm {Aut}}({\cal A}/ K(S))$ .

$(3)$ Let ${\cal K} = \{{\mathrm {Ker}} d_p: p\in I\triangleleft _m S({\cal A})\}$ and ${\cal R} = \{{\mathrm {Im}} d_p:p\in I\triangleleft _m S({\cal A})\}$ . Then the mapping $uI\mapsto (K(uI), R(uI))$ is a bijection between the Ellis subgroups of $S({\cal A})$ and the product ${\cal K}\times {\cal R}$ . Moreover, $K(uI)=K(u'I')$ iff $I=I'$ and for every $I\triangleleft _mS({\cal A})$ and $R\in {\cal R}$ there is a unique $u\in J(I)$ with $R=R(uI)$ .

We shall need a characterization of the algebras from ${\cal R}$ . We say that $A\in {\cal A}$ is generic (in G) if some finitely many left translates of A cover G. We say that a point $p\in S({\cal A})$ is generic if every set in p is generic. We say that a G-subalgebra R of ${\cal A}$ is generic if every non-empty element of R is a generic subset of G. We say that $A\in {\cal A}$ is strongly generic if it generates a generic G-subalgebra of ${\cal A}$ .

Fact 2.5 [Reference Newelski11, Reference Newelski12].

Assume ${\cal A}$ is d-closed. The G-algebras in ${\cal R}$ are exactly the maximal generic G-subalgebras of ${\cal A}$ .

The combinatorial set-up. From now on in this paper usually we assume G is an infinite group (possibly with an additional first order structure) and $L_G$ is the language of G. Also we assume that $G\prec H$ . ${\mathrm {Def}}(G)$ denotes the G-algebra of subsets of G definable by formulas of $L_G$ , using parameters from G.

Given $A\in {\mathrm {Def}}(G)$ definable by an $L_G$ -formula $\varphi (x)$ , let $A^{\#}$ be the subset of H defined by $\varphi (x)$ . Throughout, ${\cal A}$ denotes a d-closed G-algebra contained in ${\mathrm {Def}}(G)$ and ${\cal B}$ a d-closed H-algebra containing $A^{\#}$ for all $A\in {\cal A}$ . We also assume that ${\cal B}|_G:=\{B\cap G:B\in {\cal B}\}$ equals ${\cal A}$ .

Clearly, the function $\mbox {}^{\#}:{\cal A}\to {\cal B}$ mapping A to $A^{\#}$ is a Boolean algebra monomorphism respecting left translation by elements of G. So we have a dual continuous surjection $r:S({\cal B})\to S({\cal A})$ , called restriction. When $r(q)=p$ , we say that q extends p and write $p\subseteq q$ .

Remark 2.6. For every $q\in S({\cal B})$ and $A\in {\cal A}$ we have that $d_qA^{\#}\cap G=d_{r(q)}A$ .

The model theoretic set-up. This set-up appeared already in Introduction. So we have an L-structure M and $T=Th(M)$ . $G=G(M)$ is an infinite group $0$ -definable in M and we are interested in the G-flow $S_{ext,G}(M)$ of external G-types over M. We show how to reduce this set-up to the combinatorial one described above. We proceed as in [Reference Newelski10, Section 2].

For every $U\in {\mathrm {Def}}_{ext}(M)$ let $P_U$ be a new relational symbol and let $L_{ext,M}= L\cup \{P_U:U\in {\mathrm {Def}}_{ext}(M)\}$ . Let $M_{ext}$ be the expansion of M to an $L_{ext,M}$ -structure, where $P_U$ is interpreted as U.

To compare topological dynamics of flows $S_{ext,G}(M)$ for various models M of T we need to consider not just elementary, but rather $*$ -elementary extensions. We say that an L-structure N is a $*$ -elementary extension of M (and write $M\prec ^* N$ ) if $M\prec N$ and the new relational symbols of $L_{ext,M}$ are identified with some relational symbols of $L_{ext,N}$ so that $M_{ext}\prec N_{ext}|_{L_{ext,M}}$ .

It is easy to construct $*$ -elementary extensions of M. Namely, let $N^0$ be any elementary extension of $M_{ext}$ and let $N=N^0|_L$ . For $U\in {\mathrm {Def}}_{ext}(M)$ let $U^N=P_U(N^0)$ . By [Reference Newelski10, Lemma 2.1], $U^N\in {\mathrm {Def}}_{ext}(N)$ , so the identification of $P_U$ with $P_{U^N }$ witnesses that $M\prec ^* N$ .

Assume $M\prec ^* N$ and let $H=G(N)$ . Then $G\prec H$ , where G and H are considered with the structure induced from $M_{ext}$ and $N_{ext}|_{L_{ext,M}}$ , respectively. Let ${\cal A}={\mathrm {Def}}_{ext,G}(M)$ and ${\cal B}={\mathrm {Def}}_{ext,G}(N)$ . For $A\in {\cal A}$ let $A^{\#}= P_A(N)$ . By [Reference Newelski10, Lemma 1.2], $G,H,{\cal A}$ and ${\cal B}$ satisfy the assumptions of combinatorial set-up. So the results on $S({\cal A})$ and $S({\cal B})$ in the combinatorial set-up will be valid also in the model theoretic set-up.

3 Weak heirs and weak coheirs

We work in the combinatorial set-up.

Definition 3.1. Assume $p\in S({\cal A})$ and $q\in S({\cal B})$ .

$(1)\ q$ is a weak heir of p if $d_qA^{\#} = (d_pA)^{\#}$ for every $A\in {\cal A}$ .

$(2)\ q$ is a weak coheir of p if for every $A\in {\cal A}$ and $s\in S({\cal B})$ we have that

$$ \begin{align*}d_s A^{\#}\in q\iff d_sA^{\#}\cap G\in p.\end{align*} $$

In (2) “ $\iff $ ” may be equivalently replaced by “ $\Rightarrow $ ” and also by Remark 2.6, $d_sA^{\#}\cap G=d_{r(s)}A$ . In the model-theoretic set-up the definition of weak heir appeared already in [Reference Newelski10].

Lemma 3.2. Assume q is a weak heir or weak coheir of p. Then $p\subseteq q$ .

Proof Suppose that $p\not \subseteq q$ . Hence for some $A\in p$ we have that $A^{\#}\not \in q$ . Therefore $e\in d_pA$ and $e\not \in d_qA^{\#}$ . So also $e\in (d_pA)^{\#}$ and $(d_pA)^{\#}\neq d_qA^{\#}$ , showing that q is not a weak heir of p.

Let $\hat {e}=\{B\in {\cal B}: e\in {\cal B}\}$ . So $\hat {e}\in S({\cal B})$ . We have that $d_{\hat {e}}A^{\#}=A^{\#}\not \in q$ and $d_{\hat {e}}A^{\#}\cap G=d_{r(\hat {e})}A=A\in p$ , so q is not a weak coheir of p.

By Lemma 3.2, in Definition 3.1 if q is a weak heir or weak coheir of p, then $p=r(q)$ . Therefore in this situation we say also that q is a weak [co]heir over ${\cal A}$ . Regarding Definition 3.1(2), $d_{r(s)}A\in r(q)$ is equivalent to $(d_{r(s)}A)^{\#}\in q$ .

In the model theoretic set-up, when $p\in S_{ext,G}(M)$ and $q\in S_{ext,G}(N)$ is a weak [co]heir of p, we say that q is a weak [co]heir over N.

The next lemma provides an alternative definition of weak heir and weak coheir.

Lemma 3.3. Assume $p\in S({\cal A})$ and $q\in S({\cal B})$ .

$(1)\ q$ is a weak heir of p iff for every $A,B\in {\cal A}$ and $h\in H$ , if $h\in B^{\#}$ and $h^{-1}A^{\#}\in q$ , then for some $g\in G$ , $g\in B$ and $g^{-1}A\in p$ .

$(2)\ q$ is a weak coheir of p iff $p=r(q)$ and for every $A,B\in {\cal A}$ and $s\in S({\cal B})$ , if $d_sA^{\#}\cap B^{\#}\in q$ , then $d_sA^{\#}\cap B^{\#}\cap G\neq \emptyset $ .

Proof (1) $\Rightarrow $ Assume q is a weak heir of p. By Lemma 3.2, $p=r(q)$ . Take any $A,B\in {\cal A}$ and $h\in H$ satisfying $h\in B^{\#}$ and $h^{-1}A^{\#}\in q$ . Then

$$ \begin{align*}h\in d_qA^{\#}\cap B^{\#}=(d_{r(q)}A\cap B)^{\#}.\end{align*} $$

Since $G\prec H$ , there is $g\in G$ with $g\in d_{r(q)}A\cap B$ . It follows that $g\in B$ and $g^{-1}A\in r(q)$ , as required.

$\Leftarrow $ Suppose for contradiction that the right-hand side condition holds and q is not a weak heir of p. Hence $(d_pA)^{\#}\neq d_qA^{\#}$ for some $A\in {\cal A}$ . Replacing A with $G\setminus A$ if necessary, we can assume that there is $h\in d_qA^{\#}\setminus (d_pA)^{\#}$ .

Let $B=G\setminus d_pA$ . Then $h\in B^{\#}$ and $h^{-1}A^{\#}\in q$ , so by our assumptions there is $g\in G$ satisfying $g\in B$ and $g^{-1}A\in p$ . It follows that $g\in B$ and $g\in d_pA$ , a contradiction.

(2) $\Rightarrow $ Assume q is a weak coheir of p. By Lemma 3.2, $p=r(q)$ . Take any $A,B\in {\cal A}$ and $s\in S({\cal B})$ with $d_sA^{\#}\cap B^{\#}\in q$ . By the definition of weak coheir we get that

$$ \begin{align*}(d_{r(s)}A\cap B)\in p,\end{align*} $$

hence $d_{r(s)}A\cap B\neq \emptyset $ . So the right-hand side condition holds.

$\Leftarrow $ Suppose for contradiction that the right-hand side condition holds and q is not a weak coheir of $p=r(q)$ . Hence for some $A\in {\cal A}$ and $s\in S({\cal B})$ we have

$$ \begin{align*}(d_{r(s)}A)^{\#}\triangle d_sA^{\#}\in q.\end{align*} $$

Replacing A with $G\setminus A$ if necessary, we can assume that $d_sA^{\#}\setminus (d_{r(s)}A)^{\#}\in q$ . Let $B=G\setminus d_{r(s)}A$ . Then $d_sA^{\#}\cap B^{\#}\in q$ , but $d_sA^{\#}\cap B^{\#}\cap G=d_{r(s)}A\cap B=\emptyset $ , a contradiction.

We define coheirs in the standard way.

Definition 3.4. $($ 1 $)$ Let $q\in S({\cal B})$ . We say that q is a coheir over ${\cal A}$ if $B\cap G\neq \emptyset $ for every $B\in q$ .

$(2)$ For $p\in S({\cal A})$ let

$$ \begin{align*}p^{{\cal B}}=\{B\in {\cal B}: B\cap G\in p\}.\end{align*} $$

So $p^{{\cal B}}\in S({\cal B})$ is a coheir over ${\cal A}$ extending p.

Remark 3.5. Assume $q\in S({\cal B})$ . Then q is a coheir over ${\cal A}$ iff $q=p^{{\cal B}}$ , where $p=r(q)$ . In particular, every ultrafilter $p\in S({\cal A})$ has a unique coheir extension $p^{{\cal B}}$ in $S({\cal B})$ and the function $p\mapsto p^{{\cal B}}$ is a continuous embedding $S({\cal A})\to S({\cal B})$ .

So if $q\in S({\cal B})$ is a coheir over ${\cal A}$ , we say also that q is a coheir extension of $p=r(q)$ .

We can identify formulas $\varphi (x)$ of $L_G$ with parameters from H with the subsets of H they define. For $A,B\in {\cal A}$ let $\varphi _{A,B}(x,y)$ be a formula of $L_G$ (with parameters from G) saying that “ $x\in y^{-1}A\land y\in B$ ”. The condition in Lemma 3.3(1) characterizing weak heirs says that if there is $h\in H$ with $\varphi _{A,B}(x,h)\in q$ , then there is $g\in G$ with $\varphi _{A,B}(x,g)\in q$ . This shows that the notion of weak heir is indeed a weakening of the notion of heir. Similarly by Lemmas 3.5 and 3.3(2) we see that every coheir is a weak coheir. Consequently we get the following lemma.

Lemma 3.6. Every $p\in S({\cal A})$ extends to a weak heir and to a weak coheir in $S({\cal B})$ .

Proof Let $q_0=p^{{\cal B}}$ . By Remark 3.5 and Lemma 3.3(2), $q_0$ is a coheir of p, hence a weak coheir of p. Regarding the weak heir case, first extend p to a complete type $p'\in S(G)$ and let $q_1'\in S(H)$ be a heir extension of $p'$ . We can regard $q_1'$ as an ultrafilter in the algebra ${\mathrm {Def}}(H)$ . Extend $q_1'$ to $q_1"\in \beta H$ and let $q_1=q_1"\cap {\cal B}$ . By the discussion before Lemma 3.6, $q_1\in S({\cal B})$ is a weak heir of p.

Let $CH({\cal B}/{\cal A})=\{q\in S({\cal B}): q \mbox { is a coheir over }{\cal A}\}$ and similarly define the sets $WCH({\cal B}/{\cal A})$ and $WH({\cal B}/{\cal A})$ of weak coheirs and weak heirs, respectively.

Remark 3.7. $CH({\cal B}/{\cal A})$ and $WCH({\cal B}/{\cal A})$ are closed subsets of $S({\cal B})$ and $CH({\cal B}/{\cal A})\subseteq WCH({\cal B}/{\cal A})$ . The sets $CH({\cal B}/{\cal A}), WCH({\cal B}/{\cal A})$ and $WH({\cal B}/{\cal A})$ are non-empty.

Even though the set $WH({\cal B}/{\cal A})$ need not be closed (see an example at the end of this paper), it is a union of certain closed sets. For $p\in S({\cal A})$ let $WH_p({\cal B})=\{q\in WH({\cal B}/{\cal A}): p\subseteq q\}$ .

Lemma 3.8. The set $WH_p({\cal B})$ is closed and non-empty. $WH({\cal B}/{\cal A})=\bigcup _{p\in S({\cal A})}WH_p({\cal B})$ .

Proof $WH_p({\cal B})\neq \emptyset $ by Lemma 3.6. Let

$$ \begin{align*}p_{{\cal B}}=\{h^{-1}A^{\#}: h\in (d_pA)^{\#}\mbox{ and } A\in{\cal A}\}.\end{align*} $$

By Definition 3.1(1), $WH_p({\cal B})=\{q\in S({\cal B}): p_{{\cal B}}\subseteq q\}$ , so it is closed. The last part of the lemma is obvious.

We see that our definitions of weak heir and weak coheir consist in restricting the definitions of heir and coheir to some special formulas. The next proposition justifies this particular restriction.

Proposition 3.9. Assume $q\in S({\cal B})$ .

$(1)\ q$ is a weak heir over ${\cal A}$ iff $r(s*q)=r(s)*r(q)$ for every $s\in S({\cal B})$ .

$(2)\ q$ is a weak coheir over ${\cal A}$ iff $r(q*s)=r(q)*r(s)$ for every $s\in S({\cal B})$ .

Proof Note that for any $A\in {\cal A}$ and $s\in S({\cal B})$ ,

$$ \begin{align*}A\in r(s*q)\iff d_qA^{\#}\in s,\end{align*} $$
$$ \begin{align*}A\in r(s)*r(q)\iff (d_{r(q)}A)^{\#}\in s.\end{align*} $$

(1) If q is a weak heir over ${\cal A}$ , then the right-hand sides of the above equivalences are equivalent for each $A,s$ , so $r(s*q)=r(s)*r(q)$ . On the other hand, if q is not a weak heir over ${\cal A}$ , then $d_qA^{\#}\neq (d_{r(q)}A)^{\#}$ for some $A\in {\cal A}$ . Thus we can find $s\in S({\cal B})$ with $d_qA^{\#}\triangle (d_{r(q)}A)^{\#}\in s$ and then $r(s*q)\neq r(s)*r(q)$ .

(2) follows directly from definition.

In fact, Proposition 3.9(1) $\Rightarrow $ was already proved in [Reference Newelski10] in the model theoretic set-up.

For $h\in H$ let $\hat {h}=\{B\in {\cal B}: h\in H\}$ and $\hat {H}=\{\hat {h}:h\in H\}$ . Clearly $\hat {H}$ is a dense subset of $S({\cal B})$ . It is easy to see that the function $f:H\to \hat {H}$ mapping h to $\hat {h}$ is a $*$ -homomorphism, so $\hat {H}$ is a dense subgroup of $S({\cal B})$ and f is a group epimorphism. Since $*$ is a continuous in the first coordinate we get the following corollary.

Corollary 3.10. Assume $q\in S({\cal B})$ . Then q is a weak heir over ${\cal A}$ iff $r(\hat {h}*q)=r(\hat {h}) * r(q)$ for every $h\in H$ .

Notice also that $\hat {h}*q=hq$ .

Corollary 3.11. The sets $CH({\cal B}/{\cal A})\subseteq WCH({\cal B}/{\cal A})$ and $WH({\cal B}/{\cal A})$ are sub-semigroups of $S({\cal B})$ and restrictions $r:WCH({\cal B}/{\cal A})\to S({\cal A})$ and $r:WH({\cal B}/{\cal A})\to S({\cal A})$ are $*$ -epimorphisms. The restriction $r:CH({\cal B}/{\cal A})\to S({\cal A})$ is a semigroup isomorphism and homeomorphism. Its inverse is the function $j:S({\cal A})\to CH({\cal B}/{\cal A})$ mapping p to $p^{{\cal B}}$ .

Proof We prove that the set $CH({\cal B}/{\cal A})$ is closed under $*$ , the other claims of the corollary follow directly from earlier results. First, for every $q\in CH({\cal B}/{\cal A})$ and $A\in {\cal B}$ we have that

$$ \begin{align*}A\in q\iff A\cap G\in r(q)\mbox{ and } d_qA\cap G = d_{r(q)}(A\cap G).\end{align*} $$

Therefore for $p,q\in CH({\cal B}/{\cal A})$ and $A\in {\cal B}$ we have that

$$ \begin{align*}A\in p*q\iff d_qA\in p\iff d_qA\cap G\in r(p)\iff\end{align*} $$
$$ \begin{align*}d_{r(q)}(A\cap G)\in r(p)\iff A\cap G\in r(p)*r(q).\end{align*} $$

Hence $A\in p*q$ implies $A\cap G\neq \emptyset $ and $p*q\in CH({\cal B}/{\cal A})$ .

Even though the left-continuous semigroup $WH({\cal B}/{\cal A})$ may not be closed, still it has the properties from Fact 2.1 (see Proposition 4.5). In the model theoretic set-up Corollary 3.11 was already partially noticed in earlier papers. Namely in [Reference Newelski9] it was mentioned that the semigroup $S({\cal A})$ is isomorphic to $CH({\cal B}/{\cal A})$ (suitably rephrased in the model theoretic context) and in [Reference Newelski10] it was proved that $WH({\cal B}/{\cal A})$ is a semigroup and $r:WH({\cal B}/{\cal A})\to S({\cal A})$ is an epimorphism. We shall use Corollary 3.11 later to relate algebraically the Ellis groups of $S({\cal A})$ and $S({\cal B})$ . The semigroup $WH({\cal B}/{\cal A})\cap CH({\cal B}/A)$ contains an important subgroup.

Lemma 3.12. Let $\hat {G}=\{\hat {h}\in \hat {H}: h\in G\}$ .

(1) $\hat {G}$ is a subgroup of $S({\cal B})$ and the mapping $g\mapsto \hat {g}$ is a group epimorphism $G\to \hat {G}$ .

(2) $\hat {G}\subseteq WH({\cal B}/{\cal A})\cap CH({\cal B}/A)$ , $\hat {e}$ is the unique identity element in $S({\cal B})$ .

$(3)$ For every $q\in S({\cal B})$ and $g\in G$ we have that

$$ \begin{align*}q\in WH({\cal B}/{\cal A})\iff \hat{g}*q\in WH({\cal B}/{\cal A})\iff q*\hat{g}\in WH({\cal B}/A).\end{align*} $$

$(4)$ Like $(3)$ , but with $WH$ replaced by $WCH$ [or $CH$ ].

Proof The proof is by revealing the definitions and applying Proposition 3.9 in (3) and (4).

By Fact 2.3(3), elements of $S({\cal A})$ and $S({\cal B})$ may be regarded as functions from the Ellis semigroups $E(S({\cal A}))$ and $E(S({\cal B}))$ . In Proposition 3.13 we give yet another characterization of weak [co]heirs in terms of functions from $E(S({\cal A}))$ and $E(S({\cal B}))$ .

Let ${\cal A}^{\#}$ be the H-subalgebra of ${\cal B}$ generated by the set $\{ A^{\#} : A \in {\cal A} \}$ . So ${\cal A}^{\#}$ need not be d-closed. Each $q_0 \in S({\cal A})$ has a unique weak heir in $S({\cal A}^{\#})$ , that is, a type $q \in S({\cal A}^{\#})$ such that $d_q A^{\#} = (d_{q_0} A)^{\#}$ for each $A \in {\cal A}$ . We write $(q_0)^{\#} := q$ .

Consider any $p_0 \in S({\cal A}),p \in S({\cal B})$ and let $f_0 \in E(S({\cal A}))$ , $f \in E(S({\cal B}))$ correspond to $p_0$ and p via the standard isomorphisms from Fact 2.3(3). p determines a function $f^{\#}:S({\cal A}^{\#})\to S({\cal A}^{\#})$ defined by $f^{\#}(q) = p * q$ , where $p*q\in S({\cal A}^{\#})$ is defined by the formula from Fact 2.3(2). Since $p*q=\lim _{h\to p}hq$ , we have that $f^{\#}\in E(S({\cal A}^{\#}))$ and $f^{\#}$ is the unique function from $E(S({\cal A}^{\#}))$ such that the following diagram commutes.

Also using the formula for $*$ from Fact 2.3(2) we get that $d_{p*q}=d_p\circ d_q$ for $p\in S({\cal B})$ and $q\in S({\cal A}^{\#})$ .

Proposition 3.13. Assume $p_0\in S({\cal A}),p\in S({\cal B})$ and $f_0,f,f^{\#}$ are defined as above.

(1) $p$ extends $p_0$ iff the following diagram commutes.

(2) $p$ is a weak heir of $p_0$ iff the following diagram commutes.

(3) $p$ is a weak coheir of $p_0$ iff the following diagram commutes.

Proof (1) We have that

$$ \begin{align*} p \text{ extends } p_0 & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, \big( d_q A \in p_0 \Leftrightarrow (d_q A)^{\#} \in p \big) \\[1ex] & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, \big( d_q A \in p_0 \Leftrightarrow d_{q^{\#}} A^{\#} \in p \big) \\[1ex] & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, \big( A \in p_0 \ast q \Leftrightarrow A^{\#} \in p \ast q^{\#} \big) \\[1ex] & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, \big( A \in f_0(q) \Leftrightarrow A^{\#} \in f^{\#}(q^{\#}) \big) \\[1ex] & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, \big( A \in f_0(q) \Leftrightarrow A \in r(f^{\#}(q^{\#})) \big) \end{align*} $$

and the last condition means that the corresponding diagram commutes.

(2) We have that

$$ \begin{align*} p \text{ is a weak heir of } p_0 & \iff (\forall B \in {\cal A}) \, (d_{p_0} B)^{\#} = d_p B^{\#} \\[1ex] & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, (d_{p_0} d_q A)^{\#} = d_p (d_q A)^{\#} \\[1ex] & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, (d_{p_0} d_q A)^{\#} = d_p d_{q^{\#}} A^{\#} \\[1ex] & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, (d_{p_0 * q} A)^{\#} = d_{p * q^{\#}} A^{\#} \\[1ex] & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, d_{(p_0 * q)^{\#}} A^{\#} = d_{p * q^{\#}} A^{\#} \\[1ex] & \iff (\forall q \in S({\cal A}))(\forall A \in {\cal A}) \, d_{f_0(q)^{\#}} A^{\#} = d_{f^{\#}(q^{\#})} A^{\#}. \end{align*} $$

The last condition holds exactly when $f_0(q)^{\#} = f^{\#}(q^{\#})$ for each $q \in S({\cal A})$ , which is equivalent to the commutativity of the corresponding diagram.

(3) By Proposition 3.9 we have that

$$ \begin{align*} p \text{ is a weak coheir of } p_0 & \iff (\forall q \in S({\cal B})) \, r(p * q) = p_0 * r(q) \\[1ex] & \iff (\forall q \in S({\cal B})) \, r(f(q)) = f_0(r(q)), \end{align*} $$

where $r:S({\cal B})\to S({\cal A})$ is restriction. The last condition is equivalent to the commutativity of the outer rectangle in the following diagram.

Since the upper rectangle is commutative and $r : S({\cal B}) \to S({\cal A}^{\#})$ is surjective, the condition is equivalent to the commutativity of the lower rectangle.

In order to be able to compare the flows $S({\cal A})$ and $S({\cal B})$ we need a way to lift minimal left ideals in $S({\cal A})$ to minimal left ideals in $S({\cal B})$ . This uses weak heirs and was already done in [Reference Newelski10, Lemma 2.4] in the model theoretic set-up. Here we easily adapt the argument from [Reference Newelski10]. Recall from topological dynamics that a point p in a G-flow X is called almost periodic if p belongs to a minimal subflow of X, equivalently the subflow $cl(Gp)\subseteq X$ generated by p is minimal. In our situation the minimal flows in $S({\cal A})$ and $S({\cal B})$ are exactly the minimal left ideals. Their elements are exactly the almost periodic points. The next lemma is the main place where we use the assumption that $G\prec H$ in the combinatorial set-up and $M\prec ^* N$ in the model theoretic set-up.

Lemma 3.14. $(1)$ Assume $q\in S({\cal B})$ is a weak heir of $p\in S({\cal A})$ . Then $r[cl(Hq)]=cl(Gp)$ . If moreover p is almost periodic, then $r[cl(Hq')]=cl(Gp)$ for every $q'\in cl(Hq)$ .

$(2)$ Assume $I\triangleleft _mS({\cal A})$ . Then there is $I'\triangleleft _mS({\cal B})$ with $r[I']=I$ . In particular, every almost periodic $p\in S({\cal A})$ extends to an almost periodic $q\in S({\cal B})$ .

Proof (1) For $h\in H$ we have that $hq=\hat {h}*q$ . Since $*$ is left-continuous and the set $\{\hat {h}:h\in H\}$ is dense in $S({\cal B})$ , we have that $cl(Hq)=S({\cal B})*q$ . Proposition 3.9(1) implies that

$$ \begin{align*}r[cl(Hq)]= r[S({\cal B})*q]=S({\cal A})*p=cl[Gp].\end{align*} $$

In the case where p is almost periodic, $cl(Gp)$ is a minimal left ideal in $S({\cal A})$ . Since $r[cl(Hq')]$ is a G-invariant closed subset of $cl(Gp)$ , we get that $r[cl(Hq)]=cl(Gp)$ .

(2) Let $p\in I$ and let $q_0\in S({\cal B})$ be a weak heir of p. Let $I'\triangleleft _mS(B)$ be a minimal subflow of $cl(Hq_0)$ . By (1), $r[I']=I$ .

4 Transfer between $S({\cal A})$ and $S({\cal B})$ : the Ellis groups

In this section we shall relate algebraically the Ellis groups of $S({\cal A})$ and $S({\cal B})$ , under some additional assumptions. Let $I\triangleleft _mS({\cal A})$ and $u\in J(I)$ . First we lift the group $uI$ to a group in $S({\cal B})$ , possibly not an Ellis group. We do it in two ways, using the coheir and weak heir extensions.

First let ${\cal G}=j[uI]$ , where $j:S({\cal A})\to CH({\cal B}/{\cal A})$ is the $*$ -isomorphism from Corollary 3.11. So ${\cal G}$ is a subgroup of $S({\cal B})$ such that $r:{\cal G}\to uI$ is a group isomorphism. The group ${\cal G}$ is determined by the choice of $u\in J(I)$ , so we may write ${\cal G}={\cal G}_u$ . The groups ${\cal G}_u,u\in J(I),I\triangleleft _mS({\cal A})$ are all isomorphic and their choice is canonical.

Next, we find a subgroup ${\cal H}$ of $WH({\cal B}/{\cal A})$ with the property that $r:{\cal H}\to uI$ is a group epimorphism. In the model theoretic set-up this has been done in [Reference Newelski10, Section 3]. The construction from [Reference Newelski10] is easily adapted to our combinatorial set-up. We describe it briefly here, referring the reader to [Reference Newelski10, Section 3] for details.

By Lemma 3.8 and Corollary 3.11, the set $WH_u({\cal B})$ is a closed subset of $S({\cal A})$ and a sub-semigroup. Fix an $I'\triangleleft _mWH_u({\cal B})$ . By [Reference Newelski10, Lemma 3.3(2)] there is a common kernel $K'\subseteq {\cal B}$ of all $d_{q'},q'\in I'$ . Also let ${\cal R}^{\prime }_u=\{{\mathrm {Im}} d_{q'}:q'\in I'\}$ . By Fact 2.1, $I'$ is a disjoint union of isomorphic groups $u'I',u'\in J(I')$ . By [Reference Newelski10, Lemma 3.3] there is $I^+\triangleleft S({\cal B})$ with $I'=WH_u({\cal B})\cap I^+$ (in fact, $I^+=cl(Hs)$ for any $s\in I'$ ). Notice also that by Fact 3.14, $r[I^+]=I$ .

By [Reference Newelski10, Lemma 3.6], for every $q\in I$ the set $I_q':= WH_q({\cal B})\cap I^+$ is nonempty and for every $q'\in I_q'$ we have that ${\mathrm {Ker}} d_{q'}=K'$ . Also for every $q\in uI$ we have that ${\cal R}^{\prime }_u={\cal R}^{\prime }_q$ , where ${\cal R}^{\prime }_q:=\{{\mathrm {Im}} d_{q'}: q'\in I_q'\}$ . Now fix an idempotent $u'\in J(I')$ and let $R'={\mathrm {Im}} d_{u'}$ . Let

$$ \begin{align*}{\cal H}=\{q'\in\bigcup_{q\in uI}I_q': {\mathrm {Im}} d_{q'}=R'\}.\end{align*} $$

Clearly, ${\cal H}\subseteq WH({\cal B}/{\cal A})\cap I^+$ . By [Reference Newelski10, Proposition 3.7], ${\cal H}$ is a subgroup of $S({\cal B})$ and $r:{\cal H}\to uI$ is a group epimorphism, with kernel $u'I'$ . The group ${\cal H}$ is determined by the choice of $u'$ , hence we may write ${\cal H}$ as ${\cal H}_{u'}$ .

Using ${\cal H}$ , in [Reference Newelski10] we proved in the model-theoretic set-up that the Ellis groups of $S_{ext,G}(M)$ are homomorphic images of some subgroups of the Ellis groups of $S_{ext,G}(N)$ , under the assumption that there are generic types in $S_{ext,G}(N)$ (equivalently, in $S_{ext,G}(N)$ there is exactly one minimal left ideal [Reference Newelski9]). We will recall this result later in the combinatorial set-up, with an easier proof. Notice that the choice of the group ${\cal H}={\cal H}_{u'}$ is canonical. We mentioned it in [Reference Newelski10], here we make this claim precise in Proposition 4.5, that is a structural result on the semigroup $WH({\cal B}/{\cal A})$ parallel to Fact 2.1. Let $I_0^+=WH({\cal B}/{\cal A})\cap I^+$ . Hence $I_0^+\triangleleft WH({\cal B}/{\cal A})$ .

Lemma 4.1. (1) $r[I^+]=r[I_0^+]=I$ .

(2) For every $q'\in I^+_0,\ I^+=cl(Hq')$ and ${\mathrm {Ker}} d_{q'}=K'$ .

(3) If $q\in I$ , then $I_q'\neq \emptyset $ and if $q\in J(I)$ , then $I_q'\triangleleft _m WH_q({\cal B})$ .

(4) For $q\in I$ let ${\cal R}_q'=\{{\mathrm {Im}} d_{q'}:q'\in I_q'\}$ . If $u_0\in J(I)$ and $q\in u_0I$ , then ${\cal R}_q'={\cal R}_{u_0}'$ .

(5) For $v_0\in J(I_0^+)$ let

$$ \begin{align*}{\cal H}_{v_0}=\{q'\in\bigcup_{q\in u_0I}I_q': {\mathrm {Im}} d_{q'}={\mathrm {Im}} d_{v_0}\},\end{align*} $$

where $u_0=r(v_0)$ . Then $u_0\in J(I),\ {\cal H}_{v_0}\subseteq I_0^+$ is a group and $I_0^+$ is a disjoint union of the groups ${\cal H}_{v_0},v_0\in J(I_0^+)$ .

(6) If $v_0\neq v_1\in J(I_0^+)$ , then ${\mathrm {Im}} d_{v_0}\neq {\mathrm {Im}} d_{v_1}$ .

(7) The groups ${\cal H}_v,v\in J(I^+_0)$ are isomorphic and ${\cal H}_v=vI_0^+$ .

(8) $I_0^+\triangleleft _m WH({\cal B}/{\cal A})$ .

Proof (2) and (3) are [Reference Newelski10, Lemma 3.6(1),(2)]. Together with Lemma 3.14 they imply (1).

(4) is [Reference Newelski10, Lemma 3.6(3)] when $u_0=u$ . However every $u_0\in J(I)$ can play the role of u in generating $I^+$ . Namely, by (3) $I_{u_0}'\triangleleft _m WH_{u_0}({\cal B})$ and by (2), $I^+=cl(Hu_0')$ for any $u_0'\in J(I_{u_0}')$ . So the roles of u and $u_0$ in $I^+$ are symmetrical hence [Reference Newelski10, Lemma 3.6(3)] holds for arbitrary $u_0\in J(I)$ . So (4) follows.

(5) Let $v_0{\kern-1pt}\in{\kern-1pt} J(I_0^+)$ and $u_0{\kern-1pt}={\kern-1pt}r(v_0)$ . We have that $u_0{\kern-1pt}\in{\kern-1pt} J(I)$ because $r:I_0^+{\kern-1pt}\to{\kern-1pt} I$ is a $*$ -homomorphism. That ${\cal H}_{v_0}$ is a group with $r:{\cal H}_{v_0}\to u_0I$ being a group epimorphism follows from [Reference Newelski10, Proposition 3.7] and the symmetrical role of $u_0$ and u in $I^+$ .

To see that the groups ${\cal H}_{v_0},v_0\in J(I_0^+)$ are disjoint, consider $v_0\neq v_1\in J(I_0^+)$ . Let $u_i=r(v_i)\in J(I),\ i=0,1$ .

If $u_0\neq u_1$ , then $r[{\cal H}_{v_0}]=u_0I$ , $r[{\cal H}_{v_1}]=u_1I$ and $u_0I\cap u_1I=\emptyset $ , hence also ${\cal H}_{v_0}\cap {\cal H}_{v_1}=\emptyset $ .

If $u_0=u_1$ , then $v_0,v_1\in J(I_{u_0}')$ and $I_{u_0}'\triangleleft _m WH_{u_0}({\cal B})$ , so $v_0\neq v_1$ implies ${\mathrm {Im}} d_{v_0}\neq {\mathrm {Im}} d_{v_1}$ . But ${\mathrm {Im}} d_{v_i}={\mathrm {Im}} d_x$ for every $x\in {\cal H}_{v_i}$ , hence ${\cal H}_{v_0}$ and ${\cal H}_{v_1}$ are disjoint also in this case.

By (4) every $q'\in I_0^+$ belongs to some ${\cal H}_{v_0},v_0\in J(I_0^+)$ , so we are done.

(6) Assume $v_0\neq v_1\in J(I_0^+)$ . By (5) the groups ${\cal H}_{v_0}$ and ${\cal H}_{v_1}$ are disjoint and ${\mathrm {Ker}} d_{v_0}={\mathrm {Ker}} d_{v_1}=K'$ , hence by Fact 2.4, ${\mathrm {Im}} d_{v_0}\neq {\mathrm {Im}} d_{v_1}$ .

(7) Let $v_0\neq v_1\in J(I_0^+)$ . Since $K({\cal H}_{v_0})=K({\cal H}_{v_1})=K'$ , the groups ${\cal H}_{v_i}$ may be regarded as subgroups of ${\mathrm {Aut}}({\cal B}/K')$ (via the function d, since $R({\cal H}_{v_i})$ is a section of the family of $K'$ -cosets in ${\cal B}$ ). Hence the function mapping x to $v_1x$ is a group isomorphism ${\cal H}_{v_0}\to {\cal H}_{v_1}$ . We see also that ${\cal H}_{v_0}=v_0I_0^+$ .

(8) Let $x,y\in I_0^+$ . Choose $v_0,v_1\in J(I_0^+)$ such that $x\in {\cal H}_{v_0}$ and $y\in {\cal H}_{v_1}$ . It is enough to show that $y=zx$ for some $z\in I_0^+$ .

When $v_0=v_1$ , then $x,y\in {\cal H}_{v_0}$ and $z=yx^{-1}$ calculated in the group ${\cal H}_{v_0}$ will do. Otherwise, $v_1x\in {\cal H}_{v_1}$ and $z=z_0v_1$ will do, where $z_0=y(v_1x)^{-1}$ calculated in the group ${\cal H}_{v_1}$ .

Lemma 4.2. Assume $I^*\triangleleft _m WH({\cal B}/{\cal A})$ . Then $I^*$ is of the form $I_0^+$ for some $I\triangleleft _m S({\cal A}),\ u\in J(I),\ I'\triangleleft _m WH_u({\cal B})$ and $u'\in J(I')$ .

Proof Let $I=r[I^*]$ . Since $r:WH({\cal B}/{\cal A})\to S({\cal A})$ is a $*$ -epimorphism, we have that $I\triangleleft _m S({\cal A})$ . Choose $u\in J(I)$ and then $I'\subseteq I^*\cap WH_u({\cal B})$ with $I'\triangleleft _m WH_u({\cal B})$ . Let $u'\in J(I'),\ I^+=cl(Hu')$ and $I_0^+=I^+\cap WH({\cal B}/{\cal A})$ . By Lemma 4.1 and the minimality of $I^*$ we have that $I_0^+=WH({\cal B}/{\cal A})u'=I^*$ .

Lemma 4.3. Assume $I_i^+\triangleleft _m WH({\cal B}/{\cal A})$ and ${\cal R}_i=\{{\mathrm {Im}} d_{q'}:q'\in I_i^+\},i=0,1$ .

(1) If $R_0,R_1\in {\cal R}_i$ and $R_0\subseteq R_1$ , then $R_0=R_1$ .

(2) ${\cal R}_0={\cal R}_1$ .

Proof (1) Let K be the common kernel of $d_x,x\in I^+_i$ . By Fact 2.4, the algebras $R_0,R_1$ are sections of the family of K-cosets in ${\cal B}$ , so $R_0\subseteq R_1$ implies $R_0=R_1$ .

(2) Choose $v_i\in J(I_i^+),i=0,1$ . Let $x\in I_0^+$ . Hence $xv_1\in I_1^+$ and $xv_1v_0\in I_0^+$ . Therefore ${\mathrm {Im}} d_x,{\mathrm {Im}} d_{xv_1v_0}\in {\cal R}_0$ and ${\mathrm {Im}} d_{xv_1}\in {\cal R}_1$ . We have that

$$ \begin{align*}{\mathrm {Im}} d_{xv_1v_0}\subseteq{\mathrm {Im}} d_{xv_1}\subseteq {\mathrm {Im}} d_x,\end{align*} $$

hence ${\mathrm {Im}} d_{xv_1v_0}\subseteq {\mathrm {Im}} d_x$ and by (1) ${\mathrm {Im}} d_{xv_1v_0}={\mathrm {Im}} d_x$ . Therefore also ${\mathrm {Im}} d_x={\mathrm {Im}} d_{xv_1}\in {\cal R}_1$ . So ${\cal R}_0\subseteq {\cal R}_1$ . By symmetry, ${\cal R}_0={\cal R}_1$ .

Lemma 4.4. Assume for $i=0,1,\ I_i^+\triangleleft _m WH({\cal B}/{\cal A}),\ v_i\in J(I_i^+)$ and let ${\cal H}_i={\cal H}_{v_i}=v_iI_i^+$ , $I_i=r[I_i^+]\triangleleft _m S({\cal A})$ and $u_i=r(v_i)\in J(I_i)$ . Then there are group isomorphisms $f:{\cal H}_0\to {\cal H}_1$ and $\delta :u_0I_0\to u_1I_1$ such that the following diagram commutes.

Proof Let $K_i={\mathrm {Ker}} d_{v_i}$ and $R_i={\mathrm {Im}} d_{v_i}, i=0,1$ . First we consider two special cases.

Case 1. $K_0=K_1$ . In this case by Fact 2.4 and Lemma 4.1(2) we have that $cl(Hv_0)=cl(Hv_1)$ and

$$ \begin{align*}I_0^+=cl(Hv_0)\cap WH({\cal B}/{\cal A})=cl(Hv_1)\cap WH({\cal B}/{\cal A})=I_1^+.\end{align*} $$

Hence also $I_0=r[I_0^+]=r[I_1^+]=I_1$ . By Fact 2.4(2) and Lemma 4.1(5), the groups ${\cal H}_i$ are isomorphic to some subgroups of ${\mathrm {Aut}}(R_i)\cong {\mathrm {Aut}}({\cal B}/K_i)$ . Hence, considering the elements of ${\cal H}_i$ as automorphisms of ${\cal B}/K_i$ (via the function d) we see that the functions $f:{\cal H}_0\to {\cal H}_1$ and $\delta :u_0I_0\to u_1I_1$ defined by $f(x)=v_1x$ and $\delta (x)=u_1x$ are group isomorphisms and the diagram from the lemma commutes.

Case 2. $R_0=R_1$ . In this case we have that $v_0v_1=v_1$ and $v_1v_0=v_0$ . Therefore by Corollary 3.11, $u_0u_1=u_1$ and $u_1u_0=u_0$ , which gives that also ${\mathrm {Im}} d_{u_0}={\mathrm {Im}} d_{u_1}$ . Here the groups ${\cal H}_i$ are isomorphic to some subgroups of ${\mathrm {Aut}}(R_0)={Aut}(R_1)$ , the functions $f:{\cal H}_0\to {\cal H}_1$ and $\delta :u_0I_0\to u_1I_1$ defined by $f(x)=xv_1$ and $\delta (x)=xu_1$ are group isomorphisms and the diagram from the lemma commutes.

In general, by Lemma 4.3 there is $v_2\in J(I_0^+)$ with ${\mathrm {Im}} d_{v_2}=R_1$ and ${\mathrm {Ker}} d_{v_2}=K_0$ . Let ${\cal H}_2={\cal H}_{v_2}=v_2I_0^+$ and $u_2=r(v_2)\in J(I_0)$ . By Cases 1 and 2 we get group isomorphisms $f_0:{\cal H}_0\to {\cal H}_2$ , $\delta _0:u_0I_0\to u_2I_0$ , $f_1:{\cal H}_2\to {\cal H}_1$ and $\delta _1: u_2I_0\to u_1I_1$ such that the corresponding diagrams commute. Then the functions $f=f_1\circ f_0$ and $\delta =\delta _1\circ \delta _0$ satisfy our demands.

The next proposition summarizes the previous four lemmas.

Proposition 4.5. There are minimal left ideals in $WH({\cal B}/{\cal A})$ . Assume $I^*\triangleleft _m WH({\cal B}/{\cal A})$ . Then the following hold.

(1) The set $J(I^*)$ is nonempty.

(2) $I^*$ is a disjoint union of groups $u'I^*,u'\in J(I^*)$ .

(3) The groups of the form $u'I^*$ , where $I^*\triangleleft _m WH({\cal B}/{\cal A})$ and $u'\in J(I^*)$ , are isomorphic.

(4) Assume $u'\in J(I^*)$ . Then the restriction function $r:u'I^*\to uI$ is a group epimorphism, where $I=r[I^*]\triangleleft _m S({\cal A})$ and $u=r(u')\in J(I)$ . Also ${\mathrm {Ker}} (r) = I'\triangleleft _m WH({\cal B}/{\cal A})$ , where $I'=I^*\cap WH_u({\cal B})$ .

(5) The minimal ideals $I^*\triangleleft _m WH({\cal B}/{\cal A})$ are relatively closed in $WH({\cal B}/{\cal A})$ and determined by the common kernel of $d_x,x\in I^*$ .