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Many-sorted Algebras

Alberto Pardo

Grupo de Metodos Formales

Instituto de Computacion

Facultad de Ingeniera

pardo@fing.edu.uy

The aim of these brief notes is to review the fundamental concepts of many-sorted algebras.We follow the approach presented in [4]. Further details can be found in [3, 2, 1].

For any set S, we say that X is an S-sorted setif it is a family of sets indexed by S, i.e.{Xs}sS.

Definition 1 Asignature is a pairS, FwhereSis a set ofsortsand Fis a set offunctionsymbolssuch thatFis equipped with a mappingtype: F S S, which expresses thetype(or functionality) of each function symbol. One often writes f: s1 sn s to meanthat fF withtype(f) = (s1 sn,s).

Example 1 LetNat be a signature for natural numbers

Nat= {nat}, {zero, succ}where zero: nat

succ: nat nat

Definition 2 Let Xbe an S-sorted set of variables. For every sort s Swe define the setT(X)s, of terms of sort s with variables in X, as the least set containing:

(i) every variable of sorts, i.e., Xs T(X)s

(ii) every nullary function symbol (constant) c F with c: s

(iii) every termf(t1, . . . , tn) wheref: s1 sn s Fand each tiis a term inT(X)si,

i= 1, n.These sets form the family of sets {T(X)s}sS, which we shall refer to as T(X).

Note that the terms in the family {T(X)s}sS are elements (words, strings) of the set(F X {(, ),, }).

In case we want terms without variables (called ground terms) we take X=, obtainingT(), which will be written simply as T. We assume the signatures are sensiblein the sensethat each sort admits at least one ground term.

Definition 3 Given a signature = S,F, a -algebra A consists of an S-sorted familyof nonempty carrier sets{As}sSand a total function fA: As1 Asn As for each

function symbol f: s1 sn s F.

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From the family of sets of terms with variables in X, T(X), we can build a -algebra,called the algebra of terms1 and denoted by T(X), with carrier sets {T(X)s}sS and a

function fT(X)

: T(X)s1 T(X)sn T(X)s for each f: s1 sn s F suchthat

fT(X)(t1, . . . , tn)def

= f(t1, . . . , tn)

whereti T(X)si , i = 1, n. Once again, ifX= then we obtain the so-called ground termsalgebraT.

Definition 4 A -homomorphismh : A Bbetween two -algebras A and Bis a family ofmaps {hs: As Bs}sSthat preserve the operations, i.e. for each f: s1 sn s Fand ai Asi (i= 1, n),

hs(fA(a1, . . . , an)) =f

B(hs1(a1), . . . , hsn(an))

A -homomorphism that happens to be bijective is called a -isomorphism. Two -algebrasA and Bare said to be isomorphic (denoted by A=B) when there exists a -isomorphismbetween them, in other words, there are -homomorphisms h: A Band h :B A suchthat h h= idA and h h

= idB, where idA and idB are the -homomorphisms identity forthe -algebras Aand B, respectively.

Theclassof all -algebras is denotedAlg. Given a class C of -algebras, a -algebraIin C is called initialiff for every A in C there exists a unique -homomorphism h:I A.

The following proposition states that initial algebras are unique up to isomorphism.

Proposition 1

1. IfI andJare both initial in a classC of-algebras, thenI=J.

2. LetI andK be algebras inC such thatK=I. IfI is initial inC then so isK.

Proof.

1. If I and J are initial then there exist (unique) -homomorphisms h : I J andh :J Ibetween them. By composing we obtain, h h: I Iand h h :J J.But the identities idI : I I and idJ : J J are also -homomorphisms, so byuniqueness we have that h h= idI and h h

=idJ.

2. IfI is initial and K = I, then let hK : I K be the -isomorphism. For every -algebraA in C, there is an unique -homomorphismhA: I A. So,hAh

1K

:K Ais a -homomorphism as well. If g : K A is any other -homomorphism then we

can compose g hK :I A, but by initiality we have that g hK =hA and thereforeg= hA h

1K

. In summary, there exists a unique -homomorphism fromK to any otheralgebra ofC, soK is initial.

This proposition captures, in some sense, a notion of abstraction: it does not matter whatrepresentative of the class of initial algebras is being considered, it is only important the factthat it is initial.

Given an S-sorted set of variables X and a -algebra A, a map v : X A is called avaluation(or assignment) of values in A to the variables ofX. The interpretationof a termt T(X) inA (with respect to a valuation v) is a map v

:T(X) A such that:1

also called the algebra of words or Herbrand universe

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(i) v(x)def

= v(x), for each x X

(ii) v

(f(t1, . . . , tn))

def

= fA

(v

(t1), . . . , v

(tn)), for each f : s1 sn s F and ti T(X)si , i= 1, n

Proposition 2 Given a valuation v : X A, the interpretation v : T(X) A is theunique homomorphic extension ofv to T(X).

Proof. From the definitions ofT(X) andv, it turns out thatv is indeed a -homomorphism.

Hence, it remains to show that v is the unique extension of v, that is, the unique -homomorphism that satisfies the diagram,

X i T(X)

v

A

v

wherei: X T(X) is the embedding defined by i(x) =x. To prove this fact let us assumethe existence of another -homomorphism h: T(X) A which also satisfies that h i= v.We show that h= v by induction on the structure of terms.

a) for every variablex X, v(x) =v(x) =h(x) by definition.

b) for every constantc T(X),v

(c) =cA

=h(c) becausev

andhare -homomorphisms.c) for every composite term f(t1, . . . , tn) T(X)s, with f :1 sn s F and ti

T(X)si , assume as induction hypothesis that v(ti) =h(ti), for i= 1, n. Then, we can

calculate the following:

v(f(t1, . . . , tn)) = fA(v(t1), . . . , v

(tn)) (by def. v)

= fA(h(t1), . . . , h(tn)) (by hypothesis)

= h(f(t1, . . . , tn)) (h -homomorphism)

Corollary 1 There is a unique-homomorphismh: T A from the ground terms algebrato any other-algebraA. In other words,T is initial in the classAlg.

References

[1] H. Ehrig and B. Mahr. Fundamentals of algebraic Specifications 1: Equations and InitialSemantics. EATCS Monographs on Theoretical Computer Science, 6, 1985.

[2] J. A. Goguen, J. W. Thatcher, and E. G. Wagner. An Initial Approach to the Specification,Correctness and Implementation of Abstract Data Types. In R. Yeh, editor, CurrentTrends in Programming Methodology. Prentice Hall, 1978.

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[3] J. A. Goguen, J. W. Thatcher, E. G. Wagner, and J. B. Wright. Initial Algebra Semanticsand Continuous Algebras. JACM, 24(1):6895, Janeiro 1977.

[4] M. Wirsing. Algebraic Specifications: Semantics, Parameterization and Refinement. InLecture Notes of the State of the Art Seminar on Formal Description of Programming

Concepts., Petropolis, 1989. IFIP TC2 WG 2.2.

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