Examples of Algebras

4.4.1

Examples of Algebras for Basic Data Types

Let us illustrate the new definition and notations for algebras by interpreting some of the signatures for the Booleans and numbers given in Section 4.2 Our examples will be displayed rather than presented as tuples.

Booleans Given the signature ΣBooleans _{for a standard set of Boolean operations, first we}

apply the general notation for interpreting ΣBooleans_{. An arbitrary Σ}Booleans_{-algebra has the}

form:

algebra A carriers ABool

constants trueA_,falseA

: _→ABool

operations andA : ABool ×ABool →ABool

notA_{: A}

Bool →ABool

The standard interpretation of the Booleans is based on the set B=_{tt,ff_}

4.4. EXAMPLES OF ALGEBRAS 97 algebra AStandard carriers B constants tt,ff : →B operations _∧: B_×B_→B ¬: B_→B

Since tt,ff _∈B, and _∧ and _¬are functions on B, we do not have need of superscriptsB etc. Integers We introduced a signature IntegersInfix for an algebra of integers using a prefix notation for addition, subtraction and multiplication. Here is the general form of aIntegersInfix- algebra using the general notation:

algebra A carriers Aint

constants zeroA_,oneA _:

→Aint

operations addA: Aint×Aint →Aint

minusA_{: A}

int →Aint

timesA _{: A}

int×Aint →Aint

We also introduced a signatureIntegers for the integers that uses the standard infix notation for addition, subtraction and multiplication. The standard form of a Integers-algebra is based on the set

Z={. . . ,−2,−1,0,1,2, . . . ,}

of integers in decimal notation. In the standard case, we need not drop the reference to Z on the operators of the signature, say by writing + for +Z_.

algebra A carriers Z constants 0,1 : _→Z operations +Z _{: Z×Z→Z} −Z _{: Z→Z} .Z _{: Z} ×Z_→Z

Reals Here is a ΣReals_{-algebra of real numbers containing some infix and some postfix nota-}

algebra Reals carriers R,B constants 0R_{: →R} 1R_{: →R} πR _{: →R} trueR_,falseR _: →B operations +R _{: R×R→R} −R _{: R→R} .R _{: R×R→R} −1R : R_→R √ R_{: R} →R | |R_{: R→R} sinR _{: R} →R cosR _{: R→R} tanR_{: R→R} expR _{: R} →R logR _{: R} →R =R_{: R×R→B} <R_{: R×R→B} andR : B_×B_→B notR_{: B→B}

In many situations we might drop the superscript and subscript reference to R providing there could be no confusion as to which functions on reals were intended.

4.4.2

Storage Media

Thinking abstractly, an implementation of a storage medium is simply an algebra of signature ΣStorage _{which, using the general notation for interpreting signatures, has the form:}

algebra A

carriers Astore,Aaddress,Adata

constants emptyA_{: →A} store

operations inA_{: A}

data ×Aaddress×Astore →Astore

outA_{: A}

address ×Astore →Adata

4.4. EXAMPLES OF ALGEBRAS 99

algebra A

carriers Store,Address,Data constants emptyA _{: →Store}

operations inA_{: Data}

×Address_×Store _→Store outA_{: Address}

×Store _→Data

4.4.3

Machines

Thinking abstractly, an implementation of a machine is simply an algebra of signatureΣMachine_.

Using a suggestive notation for the carriers, a typical machine is an algebra of the form: algebra A

carriers State,Input,Output constants initialA : _→State

operations nextA_{: State×Input →State}

writeA_{: State}

×Input _→Output

If the set State is a finite set thenA is called a finite state machine.

4.4.4

Sets

Using the signature ΣSubsets_{-algebra of Section 4.2.2 any Σ}Subsets_{-algebra A using the general}

notation can be displayed in the following form: algebra A carriers Asubset constants emptyA _{: →A} subset universeA_: →Asubset operations unionA_{: A}

subset ×Asubset →Asubset

intersectionA_{: A}

subset ×Asubset →Asubset

complementA: Asubset →Asubset.

The intended interpretation is an algebra Ain which

(i) the carrier set Asubset is the power set P(X) of a non-empty set X, i.e., the set of all

subsets of the set X; (ii) the constants are

emptyA=∅,the empty set; universeA=X,the given set X;

(iii) the operations are defined for any V,W _{∈ P}(X) by unionA_{(V,W) =V} ∪W; intersetA_{(V,W) =V} ∩W; complementA(V) =X −V This gives us the algebra:

algebra A carriers Asubset =P(X) constants _∅: → P(X) X : _{→ P}(X) operations _∪ : _P(X)_{× P}(X)_{→ P}(X) ∩ : _P(X)_{× P}(X)_{→ P}(X) − : P(X)→ P(X)

4.4.5

Strings

Using the signatureStringsfrom Section4.2.3and applying our general mathematical notation, any ΣBasic Strings with Length_{-algebra Awill have the following mathematical form:}

algebra A

carriers Aalphabet,Astring,Anat

constants emptyA_{: →A} string

zeroA _{: →A} nat

operations prefixA : Aalphabet ×Astring →Astring

lengthA : Astring →Anat

However, when we interpret ΣBasic Strings with Length_{, we will actually have an algebra of strings}

of the form defined in Section 3.7.2, where (i) the carrier sets

Aalphabet =T, Astring =T∗, and Anat =N;

(ii) the constants

emptyA_{=², zero}A_=0;

and

(iii) the operations

emptyA_{=², prefix}A

=Prefix, lengthA=_{| |}, and succA _{= +1.}

4.5. ALGEBRAS WITH BOOLEANS AND FLAGS 101