Galois theory for sets of operations closed under permutation, cylindrification and composition

This is a postprint version of an article published asAlgebra Universalis 67(2012) 273–297.

DOI: 10.1007/s00012-012-0184-1. The final publication is available atwww.springerlink.com.

GALOIS THEORY FOR SETS OF OPERATIONS CLOSED UNDER PERMUTATION, CYLINDRIFICATION AND

COMPOSITION

MIGUEL COUCEIRO AND ERKKO LEHTONEN

Abstract. A set of operations onAis shown to be the set of linear term operations of some algebra onAif and only if it is closed under permutation of variables, addition of inessential variables and composition and it contains all projections. A Galois framework is introduced to describe the sets of oper- ations that are closed under the operations mentioned above, not necessarily containing all projections. The dual objects of this Galois connection are systems of pointed multisets, and the Galois closed sets of dual objects are de- scribed accordingly. Moreover, the closure systems associated with this Galois connection are shown to be uncountable (even if the closed sets of operations are assumed to contain all projections).

1. Introduction

Read-once functions have been widely studied in computer science, in particular, in machine learning. They are instances of linear term operations, i.e., operations induced by terms in which no variable occurs more than once. For a recent general reference, see [4].

Linear term operations of algebras are related to classes of operations that are closed under permutation of variables, addition of inessential variables and compo- sition in the same way as term operations of algebras are related to clones. Namely, we show (see Theorem 2.1) that a set of operations on a setA is the set of linear term operations of some algebra onAif and only if it is closed under the operations mentioned above and contains all projections. In other words, the sets of linear term operations of algebras on A are subuniverses of the reduct (OA;ζ, τ,∇,∗) of the full iterative algebra (OA;ζ, τ,∆,∇,∗) introduced by Mal’cev [13]. We then show that the closure system of subuniverses of (OA;ζ, τ,∇,∗) is uncountable, and we describe it in terms of a Galois connection between operations and so-called systems of pointed multisets.

This paper is structured as follows. In Section 2 we recall the notion of lin- ear term operation, and we establish a connection between linear term opera- tions and the subuniverses of the reduct (OA;ζ, τ,∇,∗) of the full iterative al- gebra (OA;ζ, τ,∆,∇,∗). We also show that the closure system of subuniverses of (O_A;ζ, τ,∇,∗) is uncountable whenever |A| ≥ 2. In Section 3, we recall the definition of Galois connection, and we discuss Galois theories for function alge- bras. In Section 4, we define a polarity between operations and systems of pointed

2000Mathematics Subject Classification. Primary: 08A40; Secondary: 06A15.

Key words and phrases. linear term operation, read-once function, function algebra, Galois connection, system of pointed multisets, permutation of variables, cylindrification, composition.

1

multisets, and we show that the Galois closed sets of operations coincide with the subuniverses of (OA;ζ, τ,∇,∗). (Section 4 of the current paper is based on [3].) In Section 5, we describe the Galois closed sets of systems of pointed multisets in terms of necessary and sufficient closure conditions. Finally, in Section 6, we locate our results among previous work done on this line of research and indicate possible directions for future work.

2. Linear term operations

Throughout this paper, we denote the set of natural numbers byω:={0,1,2, . . .} and we letAbe an arbitrary nonempty set. AnoperationonAis a mapf:Aⁿ →A for some integer n≥1, called thearity off. Forn≥1, the set of alln-ary oper- ations onA is denoted by O⁽ⁿ⁾_A , and the set of all operations onA is denoted by OA:=S

n≥1O⁽ⁿ⁾_A . For a subsetF ⊆ OAand an integern≥1, then-ary part ofF is defined asF⁽ⁿ⁾:=F ∩ O⁽ⁿ⁾_A . For each n≥1 and 1≤i≤n, then-ary operation e^n,A_i onAdefined by (a1, . . . , an)7→aiis called thei-thn-aryprojectiononA. We denote the set of all projections onAbyEA:={e^n,A_i |1≤i≤n}.

We recall basic notions related to terms and term operations, following the notation and terminology presented in [6]. For a natural number n ≥ 1, let Xn := {x1, . . . , xn} be a set of variables. Let {fi | i ∈ I} be a set of operation symbols,disjoint from the variables, and assign to each operation symbolfi a nat- ural numbern_i, called the arity of f_i. The sequence τ := (n_i)_i∈I is called atype.

Then-ary terms of typeτ are defined in the following inductive way:

(i) Every variablex_i∈X_n is ann-ary term.

(ii) If f_i is an n_i-ary operation symbol and t₁, . . . , t_ni are n-ary terms, then f_i(t₁, . . . , t_ni) is ann-ary term.

(iii) The set W_τ(X_n) of alln-ary terms is the smallest set which contains the variablesx1, . . . , xn and which is closed under the finite application of (ii).

Everyn-ary term is also anm-ary term for everym≥n. LetX :=S

n≥1Xn= {x₁, x₂, . . .}. We denote byW_τ(X) the set of all terms of typeτover the countably infinite alphabetX:

Wτ(X) := [

n≥1

Wτ(Xn).

A term islinear,if it contains no multiple occurrences of the same variable. We denote by W_τ^lin(Xn) the set of all n-ary linear terms of type τ over the alphabet Xn, and we denote byW_τ^lin(X) the set of all linear terms of type τ overX.

The number of occurrences of operation symbols in a term is called thecomplexity of the term. A terms is asubterm of a termtift=usvfor some wordsuandv.

The subterms of a linear term are linear.

LetA= (A; (f_i^A)_i∈I) be an algebra of typeτ, i.e., each fundamental operation fi has arityni, and lettbe ann-ary term of typeτoverX. The termtinduces an n-ary operationt^AonA(see Definition 5.2.1 in [6]). We call an operation induced by a linear term alinear term operation. The set of alln-ary linear term operations of the algebraA is denoted byW_τ^lin(X_n)^A, and the set of all finitary linear term operations of the algebraAis denoted byW_τ^lin(X)^A.

Mal’cev [13] introduced the operationsζ,τ, ∆,∇,∗ on the setOAof all opera- tions onA, defined as follows for arbitraryf ∈ O_A⁽ⁿ⁾,g∈ O^(m)_A :

(ζf)(x1, x2, . . . , xn) :=f(x2, x3, . . . , xn, x1), (τ f)(x1, x2, . . . , xn) :=f(x2, x1, x3, . . . , xn), (∆f)(x₁, x₂, . . . , x_n−1) :=f(x₁, x₁, x₂, . . . , x_n−1) forn >1,ζf =τ f= ∆f :=f forn= 1, and

(∇f)(x1, x2, . . . , xn+1) :=f(x2, . . . , xn+1),

(f ∗g)(x1, x2, . . . , x_m+n−1) :=f g(x1, x2, . . . , xm), xm+1, . . . , x_m+n−1 . The operations ζ and τ are collectively referred to as permutation of variables,

∆ is called identification of variables (also known asdiagonalization), ∇ is called addition of a dummy variable(orcylindrification), and∗is calledcomposition. The algebra (OA;ζ, τ,∆,∇,∗) of type (1,1,1,1,2) is called thefull iterative algebra on A, and its subalgebras are called iterative algebras on A. A subset F ⊆ OA is called aclone onA, if it is the universe of an iterative algebra onA that contains all projectionse^n,A_i , 1≤i≤n.

This paper deals mainly with the reduct (OA;ζ, τ,∇,∗) of the full iterative al- gebra onA, where the diagonalization operation ∆ is omitted. For a setF ⊆ OA, the universe of the subalgebra of (OA;ζ, τ,∇,∗) generated by F is denoted by hFi. The following theorem shows that linear terms are related to subuniverses of (O_A;ζ, τ,∇,∗) much in the same way as terms are related to clones.

Theorem 2.1. Let A= (A; (f_i^A)_i∈I) be an algebra of typeτ, and let W_τ^lin(X) be the set of all linear terms of type τ over X. Then W_τ^lin(X)^A is a subuniverse of (OA;ζ, τ,∇,∗) that contains all projections on A. Moreover, W_τ^lin(X)^A =h{f_i^A | i∈I} ∪ E_Ai.

Proof. Since for all 1 ≤i ≤n, xi ∈Xn, we have x^A_i = e^n,A_i ∈ W_τ^lin(X)^A. Thus W_τ^lin(X)^A contains all projections on A. Let f, g ∈ W_τ^lin(X)^A, say f is n-ary, g is m-ary. Then there exist linear terms t ∈ W_τ^lin(X_n), s ∈ W_τ^lin(X_m) such that t^A=f,s^A=g. Then

• t(x₂, x₃, . . . , x_n, x₁)∈W_τ^lin(X_n) andt(x₂, x₃, . . . , x_n, x₁)^A=ζf,

• t(x₂, x₁, x₃, . . . , x_n)∈W_τ^lin(X_n) andt(x₂, x₁, x₃, . . . , x_n)^A=τ f,

• t(x₂, . . . , x_n+1)∈W_τ^lin(X_n+1) andt(x₂, . . . , x_n+1)^A=∇f,

• t(s, xm+1, xm+2, . . . , x_m+n−1)∈W_τ^lin(X_m+n−1) and t(s, xm+1, xm+2, . . . , x_m+n−1)^A=f∗g.

Thus, ζf, τ f,∇f, f ∗g ∈ W_τ^lin(X)^A. Therefore, W_τ^lin(X)^A is a subuniverse of (OA;ζ, τ,∇,∗).

It is clear that {f_i^A |i ∈ I} ∪ EA ⊆W_τ^lin(X)^A, and soh{f_i^A | i∈ I} ∪ EAi ⊆ W_τ^lin(X)^A. We will show the converse inclusion by induction on the complexity of a term t. If t = xi ∈ Xn, then t^A =e^n,A_i ∈ h{f_i^A | i ∈ I} ∪ EAi. Otherwise t=fi(t1, . . . , tn_i) is a linear term and t^A∈W_τ^lin(X)^A. Then there exist numbers m1, . . . , mn_i≥1 and an injective map

σ:{(j, k)∈ω×ω|1≤j≤n_i,1≤k≤m_j} → {1, . . . , n}

such that the variables occurring in the linear term t_j (1 ≤j ≤ n_i) are precisely x_σ(j,1), . . . , x_σ(j,mj). For 1≤j≤n_i, letu_j be them_j-ary term that is obtained by

replacing the occurrence ofxσ(j,`) by x` for each 1≤`≤mj. Note that then we have

t^A_j =u^A_j(e^n,A_σ(j,1), . . . , e^n,A_σ(j,m

j)) =uj(x_σ(j,1), . . . , x_σ(j,mj))^A.

It is clear thatf_i^A ∈ h{f_i^A| i∈I} ∪ EAi, and by our induction hypothesis it also holds thatu^A₁, . . . , u^A_ni ∈ h{f_i^A|i∈I} ∪ EAi. Repeated applications ofζand∗then show that

ζ(ζ(· · ·(ζ(ζf_i^A∗u^A_ni)∗u^A_ni−1)∗ · · ·)∗u^A₂)∗u^A₁ ∈ h{f_i^A|i∈I} ∪ EAi.

Note that

ζ(ζ(· · ·(ζ(ζf_i^A∗u^A_ni)∗u^A_ni−1)∗ · · ·)∗u^A₂)∗u^A₁(a1, . . . , am₁+m₂+···+m_ni) = f_i^A(u^A₁(a1, . . . , am₁), u^A₂(am₁+1, . . . , am₁+m₂), . . . ,

u^A_ni(am1+m2+···+m_ni−1+1, . . . , am1+m2+···+m_ni)).

Furthermore, repeated applications ofζ,τ and∇ yield that then-ary function g given by

g(a1, . . . , an) =f_i^A(u^A₁(a_σ(1,1), . . . , a_σ(1,m1)), . . . , u^A_ni(a_σ(ni,1), . . . , a_σ(ni,mni))) is in h{f_i^A |i ∈I} ∪ EAi. It is obvious thatt^A=g. We conclude that{f_i^A| i∈ I} ∪ EA⊇W_τ^lin(X)^A, and the claimed equality holds.

As it is well known, there is a dichotomy of finite sets A with respect to the number of clones on A: there are a countable infinity of clones on a two-element set, whereas there are uncountably many clones on finite sets with at least three elements. As the following theorem demonstrates, there is no such dichotomy with the closure system of the subalgebras of (OA;ζ, τ,∇,∗). Of course, since clones are subuniverses of (OA;ζ, τ,∇,∗), we already know that our closure system is uncountable whenever |A| ≥ 3. Perhaps surprisingly, this is also the case when

|A|= 2.

Theorem 2.2. Let |A| ≥2.

(i) The set of subuniverses of (OA;ζ, τ,∇,∗) containing all projections is un- countable.

(ii) The set of subuniverses of (OA;ζ, τ,∇,∗)is uncountable.

In order to prove Theorem 2.2, we will make use of the following extensions of the Boolean functions introduced by Pippenger [15, Proposition 3.4]. Assume that 0 and 1 are distinct elements of A. For each integer n ≥ 3, define the function µn:Aⁿ→Aby

µn(a1, . . . , an) =







1 if (a1, . . . , an)∈ {0,1}ⁿ and

|{i∈ {1, . . . , n}:ai= 1}| ∈ {1, n−1}, 0 otherwise.

Observe thatµn(0, . . . ,0) = 0 for everyn≥3.

Lemma 2.3. LetI⊆ω\{0,1,2}andk∈ω\{0,1,2}. Thenµk ∈ h{µi|i∈I}∪EAi if and only if k∈I.

Proof. If k ∈ I, then obviously µk ∈ h{µi | i ∈ I} ∪ E_{0,1}i. Assume then that k /∈ I. Let A = (A; (µi)i∈I). By Theorem 2.1, it is enough to show that there is no linear termt of type τ := (i)i∈I such thatt^A =µk. Thus, let t be a k-ary linear term of type τ. It is clear that a k-ary linear term does not contain any operation symbols of arity greater thank, and since k /∈I, the operation symbols occurring int have arity less thank. Ift has no occurrence of operation symbols, i.e., t = x_j, then t^A = e^k,A_j . Since µ_k is not a projection, we may assume that t contains an occurrence of an operation symbol. Therefore, t has a subterm p of the form f_{`</sub>(x<sub>i1</sub>, . . . , x<sub>i`}), where ` < k and i<sub>1</sub>, . . . , i<sub>`</sub> are pairwise distinct. Let a∈ {0,1}^k be ak-tuple with 1’s at exactly`−1 positions amongi<sub>1</sub>, . . . , i<sub>`</sub> and 0’s at all remaining positions.

We will establisht^A6=µk by showing thatt^A(a)6=µk(a). Since 3≤` < k, i.e., 1 < `−1< k−1, we have µk(a) = 0. The equalityt^A(a) = 1 is entailed by the following claim.

Claim. For every subtermsoftthat containspas a subterm,s^A(a) = 1.

Proof of Claim. We first define thedepthd(s, t) of a subtermsin the linear term trecursively as follows: d(t, t) = 0; and ifs=fi(t1, . . . , tn_i) is a subterm oft with d(s, t) =d, thend(tj, t) =d+ 1 for 1≤j≤ni.

We proceed by induction on the depth dof the subterm pin s. If d = 0 then s=p, and so s^A(a) =p^A(a) =µ`(a) = 1. Assume that the claim holds ford=q for someq≥0. Let thend=q+ 1. Thens=fm(t1, . . . , tm) for somem < k, and there is ann∈ {1, . . . , m}such thatpis a subterm oft_i and the depth ofpin t_i is d. By the induction hypothesis,t^A_n(a) = 1. Furthermore, for allp6=n,

t^A_p(a) =t^A_p(0, . . . ,0) = 0,

where the first equality holds because the variables xi₁, . . . , xi_` do not occur in tp since t is a linear term. The second equality holds because the fundamental operations ofApreserve 0; hence so do all term operations onA. Thus,

s^A(a) =f_m^A(t^A₁(a), . . . , t^A_m(a)) =µm(0, . . . ,0,1,0, . . . ,0) = 1, as claimed.

The Claim implies in particular thatt^A(a) = 1. Consequently,t^A6=µ_k. Proof of Theorem 2.2. (i) By Lemma 2.3, if I, J ⊆ ω\ {0,1,2} and I 6= J, then h{µi | i ∈ I} ∪ EAi 6= h{µi | i ∈ J} ∪ EAi. Thus, there are uncountably many subuniverses of (OA;ζ, τ,∇,∗) containing all projections.

(ii) An immediate consequence of part (i).

Theorem 2.2 refers to the size of the closure system of subuniverses of (OA;ζ, τ,

∇,∗), but it does not address the lattice structure of this closure system. A con- venient way to describe closure systems is via the notion of Galois connection, which we will discuss in detail in the following section. Concerning the case of Theorem 2.2 (i), i.e., the subuniverses of (OA;ζ, τ,∇,∗) containing all projections, a Galois connection was introduced in [12], in which the defining dual objects are so-called clusters. In this paper, we extend this Galois framework to the closure sys- tem of Theorem 2.2 (ii) by considering more general dual objects, namely, systems of pointed multisets.

3. Galois connections for operations

AGalois connectionbetween setsAandBis a pair (σ, π) of mappingsσ:P(A)→ P(B) andπ:P(B)→ P(A) between the power setsP(A) andP(B) such that for allX, X⁰⊆Aand allY, Y⁰⊆B the following conditions are satisfied:

X ⊆X⁰=⇒σ(X)⊇σ(X⁰), Y ⊆Y⁰=⇒π(Y)⊇π(Y⁰), and

X⊆π(σ(X)), Y ⊆σ(π(Y)), or, equivalently,

X ⊆π(Y)⇐⇒σ(X)⊇Y.

Galois connections can be equivalently described as certain mappings induced bypolarities,i.e., relationsR⊆A×B, as the following well-known theorem shows (for early references, see [7, 14]; see also [5, 11]):

Theorem 3.1. Let A and B be nonempty sets and let R ⊆ A×B. Define the mappingsσ:P(A)→ P(B),π:P(B)→ P(A) by

σ(X) :={y∈B| ∀x∈X: (x, y)∈R}, π(Y) :={x∈A| ∀y∈Y: (x, y)∈R}.

Then the pair (σ, π)is a Galois connection between A andB.

A well-known example of a Galois connection is the Pol–Inv theory of functions and relations, which we now briefly recall. In order to facilitate the exposition, we adopt the following formalism. We regard the elements of the setω of natural numbers as ordinals, i.e.,n∈ωis the set of lesser ordinals{0,1, . . . , n−1}. Thus, an n-tuplea∈Aⁿis formally a mapa:{0,1, . . . , n−1} →A. The notation (ai|i∈n) means then-tuple mappingitoai for eachi∈n. The notation (a1, . . . , an) means then-tuple mappingito ai+1 for eachi∈n.

We view anm×nmatrixM∈A^m×n with entries inAas ann-tuple ofm-tuples M := (a¹, . . . ,aⁿ). The m-tuples a¹, . . . ,aⁿ are called the columns of M. For i∈m, then-tuple a¹(i), . . . ,aⁿ(i)

is called row i ofM. IfMi:= (aⁱ₁, . . . ,aⁱ_ni) is anm×nimatrix (1≤i≤p), then we denote by [M₁|M2| · · · |Mp] them×Pp

i=1n_i matrix

(a¹₁, . . . ,a¹_n1,a²₁, . . . ,a²_n2, . . . ,a^p₁, . . . ,a^p_np).

Anempty matrix has no columns and is denoted by ().

For a functionf:Aⁿ →B and a matrixM:= (a¹, . . . ,aⁿ)∈A^m×n, we denote byfMthem-tuple f(a¹(i), . . . ,aⁿ(i))

i∈m

inB^m, in other words,fMis the m-tuple obtained by applying f to the rows ofM.

Form≥1, we denote

R^(m)_A :={R|R⊆A^m}=P(A^m) and

RA:= [

m≥1

R^(m)_A .

Let R ∈ R^(m)_A . For a matrix M ∈ A^m×n, we write M ≺ R to mean that the columns ofMare m-tuples from the relationR. An operation f:Aⁿ →A is said to preserve R (or f is a polymorphism of R, or R is an invariant off), denoted f BR, if for all m×nmatricesM∈A^m×n,

M≺R implies fM∈R.

For a relationR∈ RA, we denote by PolRthe set of all operationsf ∈ OA that preserve the relationR. For a setR ⊆ RAof relations, we let PolR:=T

R∈RPolR.

The sets PolR and PolR are called the sets of all polymorphisms of R and R, respectively. Similarly, for an operation f ∈ OA, we denote by Invf the set of all relationsR ∈ RA that are preserved by f. For a setF ⊆ OA of functions, we let InvF:=T

f∈FInvf. The sets Invf and InvF are called the sets of allinvariants off andF, respectively.

By Theorem 3.1, (Inv,Pol) is the Galois connection induced by the polarityB between the set OA of all operations on A and the set RA of all relations on A.

It was shown by Geiger [8] and independently by Bodnarˇcuk, Kaluˇznin, Kotov and Romov [1] that for finite sets A, the closed subsets of OA under this Galois connection are exactly the clones on A. These authors also described the Galois closed subsets of RA for finite base sets A by defining an algebra on RA and showing that the closed sets of relations are exactly therelational clones, i.e., the subuniverses of the aforementioned algebra onRA.

Theorem 3.2(Geiger [8]; Bodnarˇcuk, Kaluˇznin, Kotov and Romov [1]). Let Abe a finite nonempty set.

(i) A set F ⊆ OA of operations is the set of polymorphisms of some setR ⊆ RA

of relations if and only ifF is a clone on A.

(ii) A set R ⊆ RA of relations is the set of invariants of some set F ⊆ OA of operations if and only ifRis a relational clone onA.

On arbitrary, possibly infinite setsA, the Galois closed sets of operations are the locally closed clones, as shown by Szab´o [19] and independently by P¨oschel [16]. A setF ⊆ O_Aof operations is said to belocally closed,if it holds that for allf ∈ O_A, say of arityn,f ∈ Fwhenever for all finite subsetsF⊆Aⁿ, there exists a function g∈ F⁽ⁿ⁾ such thatf|F =g|F.

These results were generalized to iterative algebras (with or without projections) by Harnau [9] who defined a polarity between operations and relation pairs. An m-aryrelation pair onAis a pair (R, R⁰) whereR, R⁰∈ R^(m)_A for somem≥1 and R⁰ ⊆R. Form≥1, denote

H^(m)_A :={(R, R⁰)|R⁰ ⊆R⊆A^m} and

HA= [

m≥1

H^(m)_A .

An operation f ∈ OA is said topreserve a relation pair (R, R⁰) ∈ H^(m)_A , denoted f B(R, R⁰), if for all matricesM∈A^m×n, the conditionM≺RimpliesfM∈R⁰. In light of Theorem 3.1, the polarity Binduces a Galois connection between the sets OA and HA. Harnau showed that the closed sets of operations are exactly the universes of iterative algebras. He defined certain operations on the set HA

of relation pairs and showed that the Galois closed subsets of relation pairs are precisely the subsets that are closed under these operations.

A Galois theory for the subuniverses of (OA;ζ, τ,∇,∗) containing all projection was proposed in [12]. Its dual objects are so-called clusters. In analogy to Harnau’s work [9], we propose in the current paper a Galois framework to characterize all subuniverses of (O_A;ζ, τ,∇,∗), not necessarily containing projections. The dual objects are so-called systems of pointed multisets, which will be defined in the following section. We will also describe the Galois closed sets of these dual objects in terms of necessary and sufficient closure conditions.

Similar studies of other reducts of the full iterative algebra (OA;ζ, τ,∆,∇,∗) have been carried out by several authors (see [1, 2, 8, 9, 10, 12, 15, 16, 19]; a brief survey of the topic is provided in [12]; see also chapter “Galois connections for operations and relations” by R. P¨oschel in [5], pp. 231–258).

4. Sets of operations closed under permutation, cylindrification and composition

We now consider the problem of characterizing the sets of operations on an arbitrary nonempty setAthat are closed under permutation of variables, addition of dummy variables, and composition (but not necessarily under identification of variables).

Afinite multiset Son a setAis a mapν_S:A→ω, called amultiplicity function, such that the set {x ∈ A | ν_S(x) 6= 0} is finite. Then the sum P

x∈Aν_S(x) is a well-defined natural number, and it is called thecardinality ofS and denoted by

|S|. The numberν_S(x) is called themultiplicity ofxinS. We may represent a finite multiset S by giving a list enclosed in set brackets, i.e.,{a1, . . . , an}, where each element x ∈A occurs νS(x) times. If S⁰ is another multiset on A corresponding to νS⁰: A → ω, then we say that S⁰ is a submultiset of S, denoted S⁰ ⊆ S, if νS⁰(x) ≤νS(x) for all x ∈ A. We denote the set of all finite multisets on A by M(A). We also denote, for each p≥0, by M^(p)(A) the set of all finite multisets onAof cardinality at mostp, i.e.,M^(p)(A) :={S∈ M(A)| |S| ≤p}.

The setM(A) is partially ordered by the multiset inclusion relation “⊆”. The join S]S⁰ and the difference S\S⁰ of multisetsS and S⁰ are determined by the multiplicity functions

νS]S⁰(x) :=νS(x) +νS⁰(x),

ν_S\S0(x) := max{νS(x)−νS⁰(x),0},

respectively. Theempty multiset onAis the zero function, and it is denoted byε.

Apartition of a finite multisetS onAis a multiset{S1, . . . , Sn} (on the set of all finite multisets onA) of nonempty finite multisets onAsuch thatS=S1] · · · ]Sn. A pointed multiset on a set A is a pair (x, S) ∈ A× M(A). The multiset {x} ]S is called the underlying multiset of (x, S). We define the cardinality of a pointed multiset (x, S) to be equal to the cardinality of its underlying multiset, i.e.,

|(x, S)|:=|{x} ]S|=|S|+ 1. If (x, S) and (x⁰, S) are pointed multisets onA, then we say that (x, S) is apointed submultiset of (x⁰, S⁰), denoted (x, S)⊆(x⁰, S⁰), if x=x⁰ andS⊆S⁰.

For an integer m ≥ 1, an m-ary system of pointed multisets on A (a system for short) is a pair (Φ,Φ⁰)∈ P(M(A^m))× P(A^m× M(A^m)), where theantecedent

Φ⊆ M(A^m) is a set of finite multisets onA^mand theconsequentΦ⁰⊆A^m×M(A^m) is a set of pointed multisets onA^m, satisfying the following two conditions:

(1) (x, S)∈Φ⁰ andS⁰⊆S imply (x, S⁰)∈Φ⁰; (2) (x, S)∈Φ⁰ implies{x} ]S ∈Φ.

For m ≥1, we denote the set of all m-ary systems of pointed multisets on A by W_A^(m), and we denote the set of all systems of pointed multisets onAby

W_A:= [

m≥1

W_A^(m).

For (Φ,Φ⁰),(Ψ,Ψ⁰)∈ P(M(A^m))× P(A^m× M(A^m)), we write (Φ,Φ⁰)⊆(Ψ,Ψ⁰) to mean componentwise inclusion, i.e., Φ⊆Ψ and Φ⁰ ⊆Ψ⁰.

For anm×nmatrixM∈A^m×n, themultiset of columns ofM is the multiset M^∗ onA^m defined by the functionχM which maps each m-tuple a ∈A^m to the number of timesaoccurs as a column of M. A matrix N∈A^m×n0 is asubmatrix ofM∈A^m×n ifN^∗⊆M^∗, i.e.,χ_N(a)≤χ_M(a) for alla∈A^m.

IfM∈A^m×nand Φ is a set of multisets overA^m, we writeM≺Φ to mean that the multisetM^∗of columns ofMis an element of Φ. IfM= (m1, . . . ,mn)∈A^m×n, n≥1, and Φ⁰ is a set of pointed multisets (x, S)∈A^m× M(A^m), then we write M≺Φ⁰to mean that (m1,{m2, . . . ,mn})∈Φ⁰. Iff ∈ O_Aⁿ and (Φ,Φ⁰)∈ W_A^(m), we say thatf preserves (Φ,Φ⁰), denoted f B(Φ,Φ⁰), if for every matrix M ∈A^m×p for somep≥0, it holds that wheneverM≺Φ andM= [M1|M2] whereM1 has ncolumns andM2may be empty, we have that [fM1|M2]≺Φ⁰.

In light of Theorem 3.1, the polarityBestablishes a Galois connection between the setsOAandWA. We say that a setF ⊆ OAof operations onAischaracterized by a setW ⊆ WA of systems of pointed multisets, if

F={f ∈ OA| ∀(Φ,Φ⁰)∈ W:f B(Φ,Φ⁰)},

i.e.,F is precisely the set of operations onAthat preserve every system of pointed multisets inW. Similarly, we say that W ischaracterized byF, if

W={(Φ,Φ⁰)∈ WA| ∀f ∈ F:f B(Φ,Φ⁰)},

i.e., W is precisely the set of systems of pointed multisets that are preserved by every operation inF. Thus, the Galois closed sets of operations (systems of pointed multisets) are exactly those that are characterized by systems of pointed multisets (operations, respectively).

Example 4.1. Let p≥ 1 be an integer and consider the set O_A^(≥p) := S

n≥pO_A⁽ⁿ⁾ of operations on A of arity at leastp. It is a subuniverse of (O_A;ζ, τ,∇,∗) which does not contain all projections. Let S be any multiset on Aof cardinality p−1.

It is easy to verify that the system ({S},∅)∈ W_A⁽¹⁾ characterizesO^(≥p)_A . For, every operation onAof arity at leastpvacuously preserves ({S},∅), but no operation of arity less thanpcan preserve ({S},∅).

Example 4.2. LetR⊆A^mbe an m-ary relation RonA. Let ΦR be the set of all finite multisetsSonA^msuch thatνS(a)6= 0 only ifa∈R, and let Φ⁰_R:=R×ΦR. It is easy to verify thatf preserves the relationRif and only iff preserves the system (ΦR,Φ⁰_R) of pointed multisets. Hence, ifC is a clone on Asuch thatC= PolRfor some setRof relations onA, thenCis characterized by the set{(ΦR,Φ⁰_R)|R∈ R}

of systems of pointed multisets.

As a corollary, for a clone C = PolR and an integer p ≥ 1, the set C^(≥p) :=

C ∩ O^(≥p)_A of at least p-ary members of C is characterized by the set {(ΦR,Φ⁰_R) | R∈ R} ∪ {({S},∅)} of systems of pointed multisets, whereS is any multiset onA with|S|=p−1.

Lemma 4.3. LetF ⊆ O_A be a locally closed set of operations that is closed under permutation of variables, addition of dummy variables, and composition. Then for every g ∈ O_A\ F, there exists a system (Φ,Φ⁰) ∈ W_A that is preserved by every operation in F but not byg.

Proof. The statement is vacuously true for F =OA, and it is easily seen to hold forF =∅. Thus, we can assume that ∅ (F (OA. Suppose that g ∈ OA\ F is n-ary. SinceFis locally closed, there is a finite subsetF⊆Aⁿ such thatg|F 6=f|F

for everyf ∈ F⁽ⁿ⁾. ClearlyF is nonempty. LetMbe a|F| ×nmatrix whose rows are the elements ofF in some fixed order.

Let µbe the smallest integer such that F^(µ)6= ∅. Note that since F is closed under addition of dummy variables, F^(m) 6= ∅ for all m ≥ µ. Let X be any submultiset of M^∗. (Recall thatM^∗ denotes the multiset of columns ofM.) Let Π := (M1, . . . ,Mq) be a sequence of submatrices of M such that {M^∗₁, . . . ,M^∗_q} is a partition of M^∗ \X where each block M^∗_i has cardinality at least µ. For 1 ≤ i ≤ q, let dⁱ ∈ FM_i, and let D := (d¹, . . . ,d^q). (Note that each FM_i is nonempty, becauseFcontains functions of arity|M^∗_i| ≥µ. Observe also that ifM_i is a submatrix ofM_j, thenFM_i⊆ FM_j, because F is closed under permutation of variables and addition of dummy variables; in particular, each FMi is a subset ofFM.) DenotedX,Π,Dc:=D^∗]X, and forX 6=M^∗, denote

hX,Π,Di:= (d¹,{d², . . . ,d^q} ]X)∈A^m× M(A^m).

Note that ifhX,Π,Di= (x, S), thendX,Π,Dc={x} ]S.

We define Φ⁰ to be the set of allhX,Π,Difor all possible choices ofX, Π, and Dsuch thatX 6=M^∗, and we define Φ to be the set of alldX,Π,Dcfor all possible choices ofX, Π andD, i.e.,

Φ :={x]S|(x, S)∈Φ⁰} ∪ {M^∗}.

We first verify that (Φ,Φ⁰) is indeed a system of pointed multisets. The first condition in the definition of a system of pointed multisets is clearly satisfied, by the definition of (Φ,Φ⁰). For the second condition, let (x, S) = hX,Π,Di ∈ Φ⁰, where Π := (M1, . . . ,Mq) and D := (d¹, . . . ,d^q). A simple inductive argument proves that (x, S⁰) ∈ Φ⁰ for all S⁰ ⊆S, and it suffices to show that (x, S⁰) ∈ Φ⁰ wheneverS=S⁰] {y} for somey∈A^m. Ify∈X, then we let

X⁰:=X\ {y}, M⁰₁:= [M1|y],

Π⁰:= (M⁰₁,M2, . . . ,Mq).

SinceF is closed under addition of dummy variables, we have thatd¹∈ FM⁰₁, and hence we have (x, S⁰) =hX⁰,Π⁰,Di ∈Φ⁰. Ify∈ {d², . . . ,d^q}, say,y=d^j, then we let

M⁰₁:= [M₁|M_j],

Π⁰:= (M₁, . . . ,M_j−1,M_j+1, . . . ,M_q),

and again, since F is closed under addition of dummy variables, we have that d¹∈ FM⁰₁ and we can let

D⁰:= (d¹, . . . ,d^j−1,d^j+1, . . . ,d^q), and we have (x, S⁰) =hX,Π⁰,D⁰i ∈Φ⁰.

Observe first thatg6B(Φ,Φ⁰). For, we have thatM≺Φ by the definition of Φ.

On the other hand, sincegM∈ F/ M, we have thatgM is not the first component ofhX,Π,Difor allX, Π, D, and hencegM⊀Φ⁰.

It remains to show that f B(Φ,Φ⁰) for allf ∈ F. Assume that f is n-ary. If N:= [N1|N2]≺Φ, whereN1hasncolumns, thenN^∗=dX,Π,Dcfor some

X, Π := (M₁, . . . ,M_q), D:= (d¹, . . . ,d^q),

where {M^∗₁, . . . ,M^∗_q} is a partition ofM^∗\X and for 1≤i ≤q, dⁱ ∈ FMi. We will show by induction onq that for everyf ∈ F, there existX⁰, Π⁰,D⁰ such that (fN1,N^∗₂) =hX⁰,Π⁰,D⁰iand hence [fN1|N2]≺Φ⁰.

Ifq= 0, then we haveX =M^∗, Π = (),D= (), anddX,Π,Dc=M^∗, and the conditionN^∗=M^∗ implies thatN1is a submatrix ofM. ThenfN1∈ FN1 and

(fN1,N^∗₂) =hM^∗\N^∗₁,(N1),(fN1)i.

Assume that the claim holds forq=k≥0, and consider the case thatq=k+ 1.

We assume thatN= [N1|N2] and N^∗=dX,Π,Dc. IfN^∗₁⊆X, thenfN1∈ FN1

and

(fN1,N^∗₂) =hX\N^∗₁,(M1, . . . ,Mk+1,N1),(d¹, . . . ,d^k+1, fN1)i.

Otherwise, for some i∈ {1, . . . , k+ 1}, dⁱ is a column of N1. Denote byN⁰₁ the matrix obtained fromN1 by deleting the columndⁱ. SinceF is closed under per- mutation of variables, there is an operationf⁰ ∈ F⁽ⁿ⁾such that fN1=f⁰[dⁱ|N⁰₁].

By the definition of dⁱ, there is an operation h∈ F such that hMi =dⁱ, and we have that

(4.1) f⁰[dⁱ|N⁰₁] =f⁰[hMi|N⁰₁] = (f⁰∗h)[Mi|N⁰₁].

SinceF is closed under composition,f⁰∗h∈ F. Furthermore, [M_i|N⁰₁|N₂]^∗=

dX]M^∗_i,(M1, . . . ,Mi−1,Mi+1, . . . ,Mk+1),

(d¹, . . . ,dⁱ⁻¹,dⁱ⁺¹, . . . ,d^k+1)c.

By the induction hypothesis, there existX⁰, Π⁰,D⁰ such that ((f⁰∗h)[M_i|N⁰₁],N₂^∗) =hX⁰,Π⁰,D⁰i.

By (4.1), we have that (f⁰∗h)[M_i|N⁰₁] =fN₁, and hence (fN₁,N^∗₂) =hX⁰,Π⁰,D⁰i.

Theorem 4.4. Let Abe an arbitrary, possibly infinite nonempty set. For any set F ⊆ OA of operations, the following two conditions are equivalent:

(i) F is locally closed and closed under permutation of variables, addition of dummy variables, and composition.

(ii) F is characterized by a setW ⊆ WA of systems of pointed multisets.

Proof. (ii)⇒(i): It is straightforward to verify that the set of operations preserving a set of systems of pointed multisets is closed under permutation of variables and addition of dummy variables. To see that it is closed under composition, let f ∈ F⁽ⁿ⁾ and g ∈ F^(p), and consider f ∗g: A^n+p−1 → A. Let (Φ,Φ⁰) ∈ W, and let M := [M₁|M2|M3] ≺ Φ, where M₁ has p columns and M₂ has n−1 columns.

SincegB(Φ,Φ⁰), we have that [gM₁|M₂|M₃]≺Φ⁰ and hence, by property (2) of the definition of a system of pointed multisets, we have [gM₁|M₂|M₃]≺Φ. Then [gM₁|M₂] hasncolumns, and sincef B(Φ,Φ⁰), we have that

f[gM₁|M₂] M₃

≺ Φ⁰. But

f[gM1|M2] = (f∗g)[M1|M2], so

(f ∗g)[M1|M2] M3

≺Φ⁰, and thusf∗gB(Φ,Φ⁰).

It remains to show thatFis locally closed. It is clear thatOAis locally closed, so we may assume thatF 6=OA. Suppose on the contrary that there is ag∈ OA\ F, say of arity n, such that for every finite subset F ⊆ Aⁿ, there is an f ∈ F⁽ⁿ⁾ such that g|F =f|F. SinceF is characterized by W andg /∈ F, there is a system (Φ,Φ⁰)∈ W such thatg6B(Φ,Φ⁰), and hence for some matrixM:= [M1|M2]≺Φ whereM1hasncolumns, we have that [gM1|M2]⊀Φ⁰. LetF be the finite set of rows of M1. By our assumption, there is an f ∈ F⁽ⁿ⁾ such that g|F = f|F, and hence

fM₁=f|_FM₁=g|_FM₁=gM₁,

and so [fM1|M2]⊀Φ⁰, which contradicts the assumption thatf B(Φ,Φ⁰).

(i) ⇒ (ii): It follows from Lemma 4.3 that for every operation g ∈ OA \ F, there exists a system (Φ,Φ⁰)∈ WA that is preserved by every operation inF but not by g. The set of all such “separating” systems of pointed multisets, for each

g∈ OA\ F, characterizesF.

5. Closure conditions for systems of pointed multisets

In this section we will describe the sets of systems of pointed multisets that are characterized by sets of operations in terms of explicit closure conditions. We will follow Couceiro and Foldes’s [2] proof techniques and adapt their notion of conjunctive minor to systems of pointed multisets. The plan is as follows. First, we introduce conditions on systems of pointed multisets, which are immediately shown to be necessary for a system’s being characterized by a set of operations. Then we prove that, when put together, these conditions are sufficient. This proof follows exactly the same general idea as in the proof of Theorem 4.4: assuming that a set W ⊆ WA fulfills the conditions, we show that for each system (Φ,Φ⁰)∈ W, there/ exists an operation that preserves every member ofWbut does not preserve (Φ,Φ⁰).

The set of all these “separating” operations, for each (Φ,Φ⁰)∈ W, constitutes a set/ of operations that characterizes W. We now introduce the technical notions and definitions needed to express these necessary and sufficient conditions.

For maps f: A → B and g: C → D, the composition g◦f is defined only if B = C. Removing this restriction, the concatenation of f and g is defined to be the map gf: f⁻¹[B ∩C] → D given by the rule (gf)(a) = g f(a)

for all a∈f⁻¹[B∩C]. Clearly, if B =C, thengf =g◦f; thus functional composition is subsumed and extended by concatenation. Concatenation is associative, i.e., for any mapsf,g,h, we haveh(gf) = (hg)f.

For a family (gi)i∈Iof mapsgi:Ai →Bisuch thatAi∩Aj=∅wheneveri6=j, we define the (piecewise)sum of the family(gi)i∈I to be the mapP

i∈Igi: S

i∈IAi→ S

i∈IBi whose restriction to each Ai coincides with gi. If I is a two-element set, say I = {1,2}, then we writeg1+g2. Clearly, this operation is associative and commutative.

Concatenation is distributive over summation, i.e., for any family (g_i)_i∈I of maps on disjoint domains and any mapf,

X

i∈I

gi

f =X

i∈I

(gif) and f X

i∈I

gi

=X

i∈I

(f gi).

In particular, ifg1andg2are maps with disjoint domains, then

(g1+g2)f = (g1f) + (g2f) and f(g1+g2) = (f g1) + (f g2).

Let g₁, . . . , g_n be maps from A to B. The n-tuple (g₁, . . . , g_n) determines a vector-valued map g: A → Bⁿ, given by g(a) := g₁(a), . . . , g_n(a)

for every a ∈ A. For f: Bⁿ → C, the composition f ◦g is a map from A to C, denoted by f(g1, . . . , gn), and called thecomposition off withg1, . . . , gn. Suppose thatA∩A⁰=

∅ and g⁰₁, . . . , g⁰_n are maps fromA⁰ to B. Let g and g⁰ be the vector-valued maps determined by (g1, . . . , gn) and (g₁⁰, . . . , g_n⁰), respectively. We have thatf(g+g⁰) = (f g) + (f g⁰), i.e.,

f (g1+g⁰₁), . . . ,(gn+g_n⁰)

=f(g1, . . . , gn) +f(g⁰₁, . . . , g⁰_n).

For B ⊆ A, ι_AB denotes the canonical injection (inclusion map) from B to A. Thus the restriction f|B of any map f: A → C to the subset B is given by f|B =f ιAB.

Remark 5.1. Observe that the notationfMintroduced in Section 3 is in accordance with the notation for concatenation of mappings. Since a matrixM:= (a¹, . . . ,aⁿ) is an n-tuple of m-tuples aⁱ: m → A, 1 ≤i ≤ n, the composition of the vector- valued map (a¹, . . . ,aⁿ) : m → Aⁿ with f: Aⁿ → B gives rise to the m-tuple f(a¹, . . . ,aⁿ) :m→B.

Letmandnbe positive integers (viewed as ordinals, i.e.,m={0, . . . , m−1}).

Leth:n→m∪V whereV is an arbitrary set of symbols disjoint from the ordinals, calledexistentially quantified indeterminate indices,or simplyindeterminates,and letσ:V →Abe any map, called aSkolem map. Then eachm-tuplea∈A^m, being a mapa:m→A, gives rise to an n-tuple (a+σ)h=: (b0, . . . , b_n−1)∈Aⁿ, where

b_i:=

(a_h(i), ifh(i)∈ {0,1, . . . , m−1}, σ(h(i)), ifh(i)∈V.

LetH := (hj)_j∈J be a nonempty family of mapshj:nj →m∪V, where each nj

is a positive integer. Then H is called a minor formation scheme with target m, indeterminate set V, andsource family (nj)_j∈J.

Let (Φj)_j∈J be a family of sets of multisets (or pointed multisets), each Φj on the setA^nj, and let Φ be a set of multisets (or pointed multisets, respectively), on A^m. We say that Φ is arestrictive conjunctive minor of the family (Φj)j∈J via H, if, for everym×nmatrixM:= (a¹, . . . ,aⁿ)∈A^m×n,

M≺Φ =⇒

∃σ1, . . . , σ_n ∈A^V ∀j∈J: (a¹+σ₁)h_j, . . . ,(aⁿ+σ_n)h_j

≺Φ_j .

On the other hand, if, for everym×nmatrixM:= (a¹, . . . ,aⁿ)∈A^m×n, ∃σ1, . . . , σn∈A^V ∀j∈J: (a¹+σ1)hj, . . . ,(aⁿ+σn)hj

≺Φj

=⇒ M≺Φ, then we say that Φ is anextensive conjunctive minor of the family (Φ_j)_j∈J via H. If Φ is both a restrictive conjunctive minor and an extensive conjunctive minor of the family (Φ_j)_j∈J viaH, i.e., for everym×nmatrixM:= (a¹, . . . ,aⁿ)∈A^m×n,

M≺Φ ⇐⇒

∃σ1, . . . , σn∈A^V ∀j∈J: (a¹+σ1)hj, . . . ,(aⁿ+σn)hj

≺Φj , then Φ is said to be atight conjunctive minor of the family (Φj)j∈J via H.

If (Φ,Φ⁰)∈ WAis a system of pointed multisets onAand (Φ_j,Φ⁰_j)_j∈J is a family of systems of pointed multisets onA(of various arities) such that Φ is a restrictive conjunctive minor of the family (Φj)_j∈J of multisets via a schemeH and Φ⁰ is an extensive conjunctive minor of the family (Φ⁰_j)_j∈J of pointed multisets via the same schemeH, then (Φ,Φ⁰) is said to be aconjunctive minor of the family (Φj,Φ⁰_j)j∈J

via H. If both Φ and Φ⁰ are tight conjunctive minors of the respective families via H, then (Φ,Φ⁰) is said to be a tight conjunctive minor of the family (Φ_j,Φ⁰_j)_j∈J via H. If the minor formation schemeH := (hj)_j∈Jand the family (Φj,Φ⁰_j)_j∈J are indexed by a singletonJ:={0}, then a tight conjunctive minor (Φ,Φ⁰) of a family consisting of a single system of pointed multisets (Φ₀,Φ⁰₀) is called asimple minor of (Φ₀,Φ⁰₀).

Lemma 5.2. Let (Φ,Φ⁰) be a conjunctive minor of a nonempty family (Φj,Φ⁰_j)_j∈J of members of WA, and let f ∈ OA. If f B (Φj,Φ⁰_j) for all j ∈ J, thenf B(Φ,Φ⁰).

Proof. Let (Φ,Φ⁰) be anm-ary conjunctive minor of the family (Φj,Φ⁰_j)j∈J via the scheme H := (h_j)_j∈J, h_j: n_j → m∪V. Let M := (a¹, . . . ,aⁿ) be an arbitrary m×n matrix such that M ≺ Φ. We want to prove that fM ≺ Φ⁰. Since Φ is a restrictive conjunctive minor of (Φj)_j∈J via H = (hj)_j∈J, there exist Skolem maps σi: V → A, 1 ≤ i ≤ n, such that for every j ∈ J, it holds that Mj :=

(a¹+σ₁)h_j, . . . ,(aⁿ+σ_n)h_j

≺Φ_j.

Since Φ⁰ is an extensive conjunctive minor of (Φ⁰_j)_j∈J via the same schemeH= (hj)_j∈J, to prove that fM≺Φ⁰, it suffices to give a Skolem mapσ:V →Asuch that, for allj ∈J, (fM+σ)hj ≺Φ⁰_j. Let σ:=f(σ1, . . . , σn). We have that, for eachj∈J,

(fM+σ)hj= f(a¹, . . . ,aⁿ) +f(σ1, . . . , σn) hj

= f(a¹+σ1, . . . ,aⁿ+σn) hj

=f (a¹+σ1)hj, . . . ,(aⁿ+σn)hj

=fMj.

By our assumptionf B(Φj,Φ⁰_j), so we havefMj≺Φ⁰_j. Let (Φ,Φ⁰),(Ψ,Ψ⁰) ∈ W_A^(m). If Φ ⊆ Ψ and Φ⁰ = Ψ⁰, then we say that (Φ,Φ⁰) is obtained from (Ψ,Ψ⁰) by restricting the antecedent. If Φ = Ψ and Φ⁰ ⊇ Ψ⁰, then we say that (Φ,Φ⁰) is obtained from (Ψ,Ψ⁰) byextending the consequent. The formation of conjunctive minors subsumes the formation of simple minors as well as the operations of restricting the antecedent and extending the consequent. Simple minors in turn subsume permutation of arguments, projection, identification of arguments, and addition of a dummy argument, operations which can be defined for

systems of pointed multisets in an analogous way as for Pippenger’s [15] constraints or Hellerstein’s [10] generalized constraints.

The m-ary trivial system of pointed multisets on A is Ωm := (M(A^m), A^m× M(A^m)). For p≥0, them-arytrivial system of pointed multisets onA of breadth pis Ω^(p)m := (M^(p)(A^m), A^m× M^(p−1)(A^m)). Them-aryempty systemonAis the pair (∅,∅). Note that Ω⁽⁰⁾m 6=∅, because ({ε},∅) is the unique member of Ω⁽⁰⁾m. The m-aryequality system onA, denotedEm, is the systemEm:= (Em, E_m⁰ ), where

Em:={S∈ M(A^m)|νS(a1, . . . , am)6= 0 =⇒ a1=· · ·=am}, E_m⁰ :={(a, . . . , a)∈A^m|a∈A} ×Em.

Clearly, every set of systems of pointed multisets characterized by a set of opera- tions contains all trivial systems, all equality systems, and all empty systems. This necessary condition can be relaxed when closure under formation of conjunctive minors is assumed.

Lemma 5.3. LetW ⊆ W_Abe a set of systems of pointed multisets that contains the binary equality system and the unary empty system. If Wis closed under formation of conjunctive minors, then it contains all trivial systems, all equality systems, and all empty systems.

Proof. The unary trivial system is a simple minor of the binary equality system via the schemeH :={h}, whereh: 2→1 is given byh(0) =h(1) = 0 (by identification of arguments). The m-ary trivial system is a simple minor of the unary trivial system via the scheme H := {h}, where h: 1 → m is given by h(0) = 0 (by addition ofm−1 dummy arguments).

For m ≥ 2, the m-ary equality system is a conjunctive minor of the binary equality system via the scheme H := (hi)_i∈m−1, where hi: 2 → m is given by hi(0) = i, hi(1) = i+ 1 (by addition of n−2 dummy arguments, restricting the antecedents and intersecting the consequents).

Them-ary empty system is a simple minor of the unary empty system via the scheme H := {h}, where h: 1 → m is given by h(0) = 0 (by addition of m−1

dummy arguments).

We define the union of systems of pointed multisets componentwise, i.e., if (Φj,Φ⁰_j)_j∈J is a family of pointed multisets on A of a common arity m, then the union of (Φ_j,Φ⁰_j)_j∈J is

[

j∈J

(Φ_j,Φ⁰_j) := [

j∈J

Φ_j,[

j∈J

Φ⁰_j .

Lemma 5.4. Let (Φj,Φ⁰_j)j∈J be a nonempty family of m-ary systems of pointed multisets onA. Iff:Aⁿ →Apreserves (Φ_j,Φ⁰_j)for every j∈J, thenf preserves S

j∈J(Φj,Φ⁰_j).

Proof. Let M := [M₁|M₂] ≺ S

j∈JΦ_j. Then M ≺ Φ_j for some j ∈ J. By the assumption that f B(Φj,Φ⁰_j), we have [fM1|M2]≺Φ⁰_j, and hence [fM1|M2]≺ S

j∈JΦ⁰_j.

Let Φ⊆ M(A^m), Φ⁰ ⊆A^m× M(A^m),S ∈ M(A^m). Thequotient of the set Φ of multisets by the multisetS is defined as

Φ/S:={S⁰∈ M(A^m)|S]S⁰ ∈Φ}.