CATEGORICAL GEOMETRY                                                                                COMMENTS TO:

Zhaohua Luo

Clones and Genoids


Universal  Algebra


Lambda Calculus


Mathematical Logic



A genoid theory is a category (A, G) of two objects such that G together with two morphisms x: G -> A and +: G -> G is the product of A and G, and G is a dense object. A left algebra of (A, G)  is a functor from (A, G)  to the category Set of sets preserving the product.

Let A = hom(G, A) and G = hom(G, G). For any pair (a, u) in A X G let [a, u]: G ->  G be the unique morphism such that x[a, u] = a and + [a, u] = u. Then u = [xu, +u] for any u in G.

A clone theory is a category (A, G) of two objects such that G together with an infinite sequence of morphisms x1, x2, .. from G to A is a countable power of A. A left algebra of (A, G)  is a functor from (A, G) to Set preserving the power. Left algebras of clone theories are main objects of study in universal algebra.

Let A = hom(G, A) and G = hom(G, G). For any infinte sequence a1, a2, .... in A  let [a1, a2...]: G ->  G be the unique morphism such that xi[a1, a2...] = ai. Then u = [x1u, x2u,...]  for any u in G. Any clone theory determines a genoid theory with x = x1, + = [ x2, x3 , ...].

Let (A, G) be a genoid. Let P be a right act of the monoid G = hom(G, G). An abstract binding (resp. cobinding) operation on P is a map σ: P ->  P such that (σp)u = σ(p[x, u+]) (resp.  σ(pu) = (σp)[x, u+]) for any p in P and u in G. A binding algebra of (A, G) is a right act P of G equipped with some binding, cobinding operations, and other homomorphism of right acts Pn -> P for various arity n 0. Here are two important types of binding algebras:

Lambda Calculus: A reflexive lambda calculus is a genoid (A, G) together with a binding operation λ: A ->  A and a cobinding operation λ*: A ->   A such that λλ*  = idA. An extensive lambda calculus is a genoid (A, G) together with a bijective binding operation λ: A ->  A.

First Order Logic: A quantifier algebra with terms in a genoid (A, G)  is a right act P of G with an abstract binding operation : P ->  P and two homomorphisms F: P0 -> P, and =>: P X P -> P such that
(i)  P is a Boolean algebra with respect to the derived operations (V, Λ, ~, F, T).
(ii) ∃ (p q) = (∃ p) (∃ q) for any p,q, in P.
(iii)  p < (p)+ for any p in P.


A genoid (A, G, x, +, [ ]) consisting of a monoid (G, e), a right act A of G, an element x of A, an element + of G, and a map [  ]: A X G -> G such that for any a in A and u in G we have
(i)  x[a, u] = a.
(ii) +[a, u] = u.

(iii) u =  [xu, +u].
In the following we fix a genoid, denoted simply by (A, G). We shall write [a1, a2,, u]  for [a1[a2, [... [an, u]...]]].

Let x1 = x, xi =   xi+  for any i > 0. Then (iii) extends to

u  = = [x1u, x2u, ... xnu, +nu]
for any n > 0. Let κn: G -> An be the map sending each u to the expression [x1u, x2u, ... xnu].  We assume y, z, w, {x1, x2 , x3 ...}, which are called syntactical variables.
Let P be a right act of the monoid G. We say an element p of P has a finite rank n > 0 if pu = pv for any u, v in G with κn(u) = κn(v). We say p has a finite rank 0 (or p is closed) if pu = pv for any u, v in G. An element of P is called finitary if it has a finite rank. We say P (or A) is locally finitary if any of its element is finitary. We say an element p of P is independent of  xn if p = p[x1, .. x(n-1), x(n+1), +n].

Let  δ: G -> G be the map sending u to [x, u+]. Then δ is an endomorphism of monoid G. Let - = [x, e], where e is the unit of G. Then (δ, +, -) is a monad on G (viewed as a one-object category). Thus G is a Kleisli algebra.

If P is any right act of G denote by PA  the new right act  (P, *) of G defined by p*u = p(δu) = p[x, u+]. A map σ: P -> P is an abstract binding (resp. cobinding) operation if it is a homomorphism from PA to P (resp. from P to PA). Such abstract binding operations can be applied to defined algebraic theories for lambda calculus and first order logic. The traditional binding operation σxi : P -> P (for each variable xi) is then defined as the derived operation:
σxi.p = σ(p[x2, x3, ..., xi, x1, +i+1]).

If y = xi then  σy.p means σxi.p.

The main difference between an abstract binding σ: P -> P  and a derived operation σy: P -> P is that σ reduces the rank n0 of a finitary element of P by 1 while σy makes  any element of P independent of the variable y (i.e. binding y). Hence if p has a finite rank n > 0 then σnp and σx1. ... σxn.p  are always closed.
If P is a right act of G let ev: PA X A -> P be the map defined by ev(p, a) = p[a, e].  For any right act T of G define  Λ: hom(T X A, P) -> hom(TPA) given by

f)t = f(t+, x).

Define  Λ-1: hom(TPA) ->  hom(T X A, P)  by

-1g)(t, a) = g(t)[a, e] = ev(g(t), a).

It is easy to see that  Λ  is the inverse of Λ-1. Hence Λ is bijective. Thus (PA, ev) is the exponent in the the cartesian closed category ActG of right acts of G.

Letting T = P = A  we obtain a bijection Λ: hom(A X AA)-> hom(AAA).  This is the starting point of lambda calculus.
We say (A, G) is an extensive genoid (or (extensive) lambda calculs) if it has a bijective abstract binding operation λ: A -> A. Denote by λ* the inverse of λ. Then λ* is a cobinding operation. Thus λ* determines a homomorphism Λ-1(λ*): A X A -> A defined by

ab = λ*a[b, e].

Hence a genoid is extensive iff there is a homomorphism A X A -> A of right acts of G such that the induced cobinding operation λ* defined by

λ*(a)  = (a+)x

is bijective. Since  idA = λ*λ  = λλ*, we have the following two axioms

a = (λa)+x (or a = (λy.a)y for any y) (β-conversion)


a = λ(a+x) (or a = λy.(ay) if a is independent of y) (η-conversion)


We list some of the useful properties for lambda calculus: (a, b, c in A and u in G)

(1) (λa)b = a[b, e].

(2) ((
λ a)u)b = a[b, u].

(3) (
λa+)b = a.

(4) (
λn a)+n xn...x1 = a for any integer n > 0.

(5) If a has a finite rank n > 0 then (
λna)xn...x1 = a and λna is closed. Thus

(λna)an...a1 = (λna)xn...x1 [a1, ..., an, e]  = a[a1, ..., an]

(6) An element a has a finite rank n > 0 if and only if there is a closed element c such that

a = cxn...x1.

The following closed terms play important role in lambda calculus (notation: λy1...yn.a = λy1.y2.(..(λyn.a)...)).)

I = λy,y =  λx1.
K = λyz.y = λλ x2
S = λyzw.yw(zw) = λλλx3x1(x2 x1).

It follows from (5) we have

Iax1 [a, e] = a.

Kab = x2[b, a, e] = a.

Sabc = (x3x1(x2x1))[c, b, a, e] = ac(bc).



A clone is a set A such that
(i) The set A* of all the infinite sequences [a1, a2, ...] of elements of A is a monoid with a unit [x1, x2, ...].
(ii) A is a right act of A*.
(iii)  xi[a1, a2, ...] = ai for any [a1, a2, ...] and i > 0.
Any clone A determines a genoid (A, A*, x1, +, [ ]) with + = [x2, x3, ...] and [a1, [b1, b2 ,..]] = [a1, b1, b2 ,..]. Conversely, assume (A, G) is a any genoid. Denote by F(A) the set of finitary elements of A. For any a in F(A) with a finite rank n > 0 and [a1, a2, ...]  in F(A)* define a[a1, a2, ...] = a[a1, a2, ..., an , e], which is independent of the choice of n. Let 

[a1, a2, ...][b1, b2, ...] = [a1[b1, b2, ...], a2[b1, b2, ...], ...]

Then F(A)* is a monoid with the unit [x1, x2, ...], F(A) is a right act of A*, and xi[a1, a2, ...] = ai. Thus (F(A), F(A)*) is a locally finitary genoid. If A = F(A) is locally finitary then we have a canonical homomorphism of genoids (A, G) -> (A, A*) sending each u in G to [x1u, x2u, ... xnu, ...]. Since any abstract binding operation preserving finitary elements, if (A, G) is a lambda calculus, then so is (F(A), F(A)*).

The class of genoids forms a variety of 2-sorted heterogeneous finitary algebras with universes A and G.  The class of clones forms a variety of infinitary algebras with universe A.
Let N be a full subcategory of a category X.  A Kleisli triple over N is a system T = (T, η,
*-) consisting of functions
(a) T: Ob N
 ->  Ob X,
(b) η assigns to each object A in N a morphism ηA: A
 -> TA,
(c) *- maps each morphism f: B -> TC with B, C in N to a morphism *f: TB
-> TC, such that for any g: C -> TD with D in N
(i) *f*g = *(f*g ).
(ii)  ηB*f = f.
(iii) *ηC = idTC.
If N = X we obtain the original definition for a Kleisli triple over a category, which is an alternative description of a monad.
Let N be a full subcategory of a category X. A clone over N is a pair (K, T) where K is a category and T is a functor T: K -> X such that for any A, B, C in N
(i). Ob N = Ob K.
(ii). K(A, B) = X(A, TB).
(iii). f(Tg) = fg for any f in K(A, B) and g in K(B, C).
The notions of a clone over N is equivalent to a Kleisli triple over N defined above.. 
Examples: Let X = Set be the category of sets.  
1. A clone over a singleton is equivalent to a monoid.
2. A clone over a finite set is equivalent to a unitary Menger algebra.
3. A clone over a countable set is  equivalent to a clone defined above.
4. A clone over the subcategory {0, 1, 2, ...} of finite sets  is equivalent to a clone in the classical sense (or a Lawvere theory).
5. A clone over a one-object category is equivalent to a Kleisli algebra.


1. Clone Theory, its Syntax and Semantics, and Applications to Lambda Calculus and Algebraic Logic pdf
2. Clones and Genoids (Basic Concepts) html pdf
3. Clones and Genoids in Lambda Calculus and First Order Logic   pdf

Mathematics Books | Universal Algebra | Lambda Calculus | Logic | Category Theory

Created: Dec 1 2007
Comments to:
Copyright 2007  by Zhaohua Luo  All rights reserved

1> 0 <1