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$\Sigma$-Algebra
  
 
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Formulas are syntactically correct expressions in a formalized language defined over a signature, a set of variables, and a logics. In this way, formulas are quite similar to terms. Since predicates and logics symbols are included in their inductive definition, they represent truth values instead of sort values.
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$\Sigma$-Algebras are the [[Semantics|semantical]] counterpart to the [[Signature (Computer Science)|signatures]], which are pure syntactical objects. In order to give the function symbols $f\in F$ of a signature $\Sigma=(S,F)$ a meaning, a (total) $\Sigma$-algebra provides an object with the same structure as $\Sigma$ but consisting of concrete elements and concrete functions operating on these elements. The elements and functions of a $\Sigma$-algebra are the counterparts to the sorts and function symbols of the signature $\Sigma$.
 
 
For examples of the exact definition of the concept of a formula in several formalized languages, see the articles [[Axiomatic set theory|Axiomatic set theory]]; [[Arithmetic, formal|Arithmetic, formal]]; [[Predicate calculus|Predicate calculus]]; [[Types, theory of|Types, theory of]]. In mathematical practice, formulas also have a semantic meaning. They can be either names, or forms of statements, definition-abbreviations, etc.
 
 
 
===Definition of Formulas===
 
  
Let $\Sigma =(S,F)$ be a signature and $P$ be a set of predicate symbols for $S$ with range $\mathbb{B}$ representing the set of truth values of the underlying logics. As usual, it holds $P\cap S=\emptyset$ and $P\cap F= \emptyset$. The notions of arity and type defined for the function symbols $f\in F$ may also be defined for the predicate symbols $p\in P$: Every $p\in P$ is assigned an <i>arity</i> ar$\colon P\longrightarrow \mathbb{N}_0$ giving the number of arguments of $p$. Every predicate symbol $p\in P$ is also assigned a <i>predicate type</i> type$\colon s_1\times\cdots\times s_{ar(p)} \longrightarrow \mathbb{B}$.
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===Definition of $\Sigma$-Algebras===
  
Let $X_s$ be a set of free variables of sort $s\in S$ with $X_s\cap F=\emptyset$, $X_s\cap P=\emptyset$, and $X_s\cap S=\emptyset$. Furthermore, let the set of free variables be defined as disjoint union $X:= \bigcup_{s\in S} X_s$. Then the set $Q(\Sigma,P,X)$ of <i>atomic formulas</i> consists of all $p(t_1,\ldots,t_n)$ for predicates $p\in P$ with type$(p)= s_1\times\cdots\times s_{ar(p)}$ and $t_i\in T_{s_i}(\Sigma,X)$.  Examples for such predicates $p\in P$ are properties, equalities, inequalities etc. Atomic formulas are also called <i>atoms</i>.
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Formally, a $\Sigma$-algebra $A=((s^A)_{s\in S},(f^A)_{f\in F})$ consists of a family $(s^A)_{s\in S}$ of <i>carrier sets</i> $s^A$ corresponding to the sorts $s\in S$ and a family $(f^A)_{f\in F}$ of functions on these carrier sets corresponding to the function symbols $f\in F$. The compatibility requirement is that for a function symbol $f$ of type$(f)= s_1\times\cdots\times s_n \longrightarrow s$, the function $f^A$ must have the form $f^A\colon s_1^A\times\cdots\times s_n^A \longrightarrow s^A$.
  
The set $L(\Sigma,P,X)$ of (general) <i>formulas</i> depends on the underlying logics. For PL1, it is the smallest set containing
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===Category of $\Sigma$-Algebras===
* the atomic formulas $Q(\Sigma,P,X)$
 
* the expressions $p_1\vee p_2$, $p_1\wedge p_2$, $\neg p$ for $p,p_1,p_2\in L(\Sigma,P,X)$
 
* the expressions $\forall x_1\in X_{s_1},\ldots,x_n\in X_{s_n}\colon p(x_1,\ldots,x_n)$ and $\exists x_1\in X_{s_1},\ldots,x_n\in X_{s_n}\colon p(x_1,\ldots,x_n)$ for predicates $p\in P$ with type$(p)= s_1\times\cdots\times s_{ar(p)}$
 
  
===Identifying and Manipulating Free Variables===
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Let $\Sigma=(S,F)$ be a signature and let $A,B$ be $\Sigma$-algebras.  A $\Sigma$-algebra-morphism $m\colon A\longrightarrow B$ is a family $(m_s\colon s^A \longrightarrow s^B)_{s\in S}$ of mappings between the carrier sets of $A,B$ fulfilling the following compatibility properties
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* $m_s(f^A)= f^B$ for $f\in F$ with ar$(f)=0$ and $\mathrm{type}(f)=\,\, \rightarrow s\}$
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* $m_s(f^A(a_1,\ldots,a_n))=f^B(m_{s_1}(a_1),\ldots,m_{s_n}(a_n))$ for $f\in F$ with $\mathrm{type}(f)= s_1\times\cdots\times s_n \longrightarrow s$ and for $a_i\in s_i^A$.
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The class of $\Sigma$-algebras together with the $\Sigma$-algebra-morphisms forms a [[Category|category]] {{Cite|W90}}.
  
Analogous to terms, the existence or nonexistence of free variables in a formula makes a fundamental difference (see for example section [[Formula#Sentences and Atomic Formulas]]. Thus, procedures for determining and manipulating free variables in formulas exist corresponding to the ones defined for terms. These procedure are somewhat more complicated as in the case of terms, however, for handling the additional logics symbols and the existence of bound variables.
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The option to use partially defined functions complicates the situation considerably and leads to refined versions of the definition. Though this does not belong to the scope of this entry, a short remarks seems to be appropriate. For example it is possible under this generalization that $m_s(f^A(a_1,\ldots,a_n))$ is undefined though $f^A(a_1,\ldots,a_n)$ is defined. This can be caused by an undefinedness of a term $m_{s_j}(a_j)$ or of $f^B(m_{s_1}(a_1),\ldots,m_{s_n}(a_n))$ {{Cite|M89}}.
  
A mapping $V\colon L(\Sigma,P,X) \longrightarrow 2^X$ for identifying the free variables in a formula is inductively defined as follows:
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===$\Sigma$-Subalgebras===
* For a predicate $p(t_1,\ldots,t_n)$ with $p\in P$ of type$(p)= s_1\times\cdots\times s_{ar(p)} $ and terms $t_i\in T_{s_i}(\Sigma,X)$, it holds $V(p(t_1,\ldots,t_n)) := V(t_1)\cup\cdots\cup V(t_n)$
 
* For a formula $p\in L(\Sigma,P, X)$, it holds $V(\neg p) = V(p)$
 
* For a formula $p_1,p_2\in L(\Sigma,P, X)$, it holds $V(p_1\vee p_2) = V(p_1)\cup V(p_2)$ and $V(p_1\wedge p_2) = V(p_1)\cup V(p_2)$.
 
* For a formula $p\in L(\Sigma,P, X)$, it holds $V(\forall x\colon p) = V(p)\setminus \{x\}$ and $V(\exists x\colon p) = V(p)\setminus \{x\}$
 
  
Let $p\in L(\Sigma,P,X)$ be a formula, $w\in T(\Sigma,X)$ be a term, and $x\in X$ be a variable.  The <i>substitution</i> $p[x\leftarrow w]$ of $x$ with $w$ is inductively defined as follows:
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One can easily imagine that typically many different $\Sigma$-algebras exist for the same signature $\Sigma$. This holds even in the case of [[Algebraic Specification|algebraic specifications]], which can restrict the set of admissable $\Sigma$-algebras by additional [[Axiom|axioms]]. In effect, this means that an abstract signature $\Sigma$ can be 'implemented' by concrete $\Sigma$-algebras with different semantics. This leads to an interest in the relationships between these $\Sigma$-algebras. One method to discuss the relationships is to use $\Sigma$-algebra-morphisms, another is the notion of $\Sigma$-subalgebras.
* $x[x\leftarrow w]:= w$
 
* $y[x\leftarrow w]:= y$ for $y\in X$ with $x\neq y$
 
* $p(t_1,\ldots,t_n)[x\leftarrow w] := p(t_1[x\leftarrow w],\ldots,t_n[x\leftarrow w])$ for a predicate $p\in P$ of type$(p)= s_1\times\cdots\times s_{ar(p)} $ and terms $t_i\in T_{s_i}(\Sigma,X)$.
 
* $(\neg p)[x\leftarrow w] = \neg (p[x\leftarrow w])$ for a formula $p\in L(\Sigma,P, X)$
 
* $(p_1\vee p_2)[x\leftarrow w] = p_1[x\leftarrow w]\vee p_2[x\leftarrow w]$ and $(p_1\wedge p_2)[x\leftarrow w] = p_1[x\leftarrow w]\wedge p_2[x\leftarrow w]$ for formulas $p_1,p_2\in L(\Sigma,P, X)$
 
* $(\forall y\colon p)[x\leftarrow w] = \forall y\colon p[x\leftarrow w]$ and $(\exists y\colon p)[x\leftarrow w] = \exists y\colon p[x\leftarrow w]$ for a formula $p\in L(\Sigma,P, X)$ and a variable $y\in X$, $y\neq x$
 
* $(\forall x\colon p)[x\leftarrow w] = \forall x\colon p$ and $(\exists x\colon p)[x\leftarrow w] = \exists x\colon p$ for a formula $p\in L(\Sigma,P, X)$
 
The last rule of the definition holds, since $x$ is already used as a bound variable in the quantified expressions. Multiple substitutions $p[x_1\leftarrow w_1,\ldots, x_n\leftarrow w_n]$ can be defined analogously. One has to note, that the substitutions of $x_1,\ldots,x_n$ with $w_1,\ldots,w_n$ are executed simultaneously and not consecutively.  Otherwise, in the case of $x_j\in V(w_k)$, different results have to be expected.
 
  
===Sentences and Atomic Formulas===
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$\Sigma$-Subalgebras of $\Sigma$-algebras are defined in the usual way.  A $\Sigma$-Algebra $A$ is called a <i>$\Sigma$-subalgebra</i> of a $\Sigma$-Algebra $B$, if $\forall s\in S\colon s^A\subseteq s^B$ and if $f^A(a_1,\ldots,a_n) = f^B(a_1,\ldots,a_n)$ for $f\in F$ with $\mathrm{type}(f)= s_1\times\cdots\times s_n \longrightarrow s$ and for $a_i\in s_i^A$. The subalgebra-property is written as $A\subseteq B$.
  
A formula $p\in L(\Sigma,P,X)$ is called <i>closed</i>, if $V(p)= \emptyset$. Closed formulas are also called <i>sentences</i>. They do not necessarily belong to the set $L(\Sigma,P,\emptyset)$, however, because bound variables can exist in $p$ (e.g. quantifier variables).  Sentences may assigned a fixed truth value contrary to general formulas.  Since one can instantiate free variables with any truth value, the truth value of a general formula can vary according to the specific instantiation as well.
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Another way to characterize the subalgebra-property is the existence of a $\Sigma$-algebra-morphism $m\colon A\longrightarrow B$, which is the identity on each carrier set of $A$, i.e. $m=(\mathrm{id}_{s^A})_{s\in S}$.
  
An atom $p(t_1,\ldots,t_n)\in Q(\Sigma,P,X)$ is called a <i>ground atom</i>, if the terms $t_i$ do not contain a free variable (i.e. if $V(t_i)=\emptyset$ for $i=1,\ldots,n$). The ground atoms are the closed atomic formulas. They are also called <i>atomic sentences</i>.  Sentences are defined inductively based on atomic sentences by application of connective and quantifier symbols.  
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(Carriers of) $\Sigma$-Subalgebras are closed under [[Intersection of sets|intersection]]. For a family $((s^{A_i})_{s\in S},(f^{A_i})_{f\in F})_{i\in I}$ of $\Sigma$-subalgebras $A_i$, their intersection $A=((s^{A})_{s\in S}, (f^{A})_{f\in F})$ is a $\Sigma$-subalgebra given by carrier sets $s^A:= \bigcap\limits_{i\in I} s^{A_i}$ for all $s\in S$ and functions $f^A:= f^{A_k}|_{s^A}$ for all $f\in F$ for an arbitrarily chosen $k\in I$. The declaration $f^A$ is well-defined, because according to the definition of a $\Sigma$-subalgebra the functions $f^{A_i}$ must behave in the same way on $s^A$.
  
===Morphisms===
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The closedness of the set of $\Sigma$-subalgebras under intersections has an important consequence. For a $\Sigma$-algebra $A=((s^A)_{s\in S}, (f^A)_{f\in F})$ and an $S$-sorted set $X=(\bar s)_{s\in S}$ with $\bar s\subseteq s^A$ for all $s\in S$, it assures the existence of a smallest $\Sigma$-subalgebra $A'\subseteq A$ of $A$ containing $X$, i.e.  $\bar s\subseteq s^{A'}$. The $\Sigma$-algebra $A'$ is called the $\Sigma$-subalgebra of $A$ <i>generated</i> by $X$ {{Cite|ST99}}.
  
A signature morphism $m\colon \Sigma_1\longrightarrow \Sigma_2$ can be extended to a morphism between formulas, if some weak additional assumptions are fulfilled. This shows that basically the existence of a signature morphism suffices to relate the formalized languages of formulas (and terms) defined over $\Sigma_1$ and $\Sigma_2$ as well.  The extension of $m$ is composed of the following parts:
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The subalgebra-property is compatible with $\Sigma$-algebra-morphisms.  Let $A,B$ be $\Sigma$-algebras and let $m=(m_s)_{s\in S}\colon A \longrightarrow B$ be a $\Sigma$-algebra-morphism. For a $\Sigma$-subalgebra $A'\subseteq A$ of $A$, its image $m(A')\subseteq B$ is a $\Sigma$-subalgebra of $B$.  The expression $B':=m(A')$ is defined in the obvious way, i.e. for $A'=((s^{A'})_{s\in S},(f^{A'})_{f\in F})$ the $\Sigma$-subalgebra $B'=((s^{B'})_{s\in S},(f^{B'})_{f\in F})$ is given by $s^{B'}:=\{m_s(a)|a\in s^{A'}\}$ for all $s\in S$ and $f_{B'}(m_{s_1}(a_1),\ldots,m_{s_n}(a_n)) = m_s(f_{A'}(a_1,\ldots,a_n))$ for all function symbols $f \colon s_1 \times \ldots\times s_n \longrightarrow s$ with $f\in F$ and $a_i \in s_i^{A'}$. The coimage of a $\Sigma$-subalgebra $B'\subseteq B$ of $B$ under $m$ is a $\Sigma$-subalgebra $m^{-1}(B')\subseteq A$ of $A$. The expression $m^{-1}(B')$ can be defined analogously to $m(A')$ and is omitted here {{Cite|ST99}}.
* For the terms contained in a formula, the corresponding extension of $\sigma$ to terms have to be used.
 
* For predicates, some kind of compatibility is required.
 
* Connectives and quantifiers remain unchanged.
 
Formally, this leads to the following statement: Let $m\colon \Sigma_1\longrightarrow \Sigma_2$ be a signature morphism for signatures $\Sigma_1=(S_1,F_1), \Sigma_2=(S_2,F_2)$.  The morphism $m$ may be extended to a mapping, which is defined for sets $X = \bigcup_{s\in S_1} X_s$, $X'= \bigcup_{s\in S_2} X_s'$ of variables as well with $m(x)\in X'_{m(s)}$ for $x\in X_s$, $s\in S_1$.  Furthermore, the extension may be defined also for predicates $P_1$ defined over $S_1$ and $P_2$ defined over $S_2$ such that the predicate type is preserved under $m$ according to $\forall p\in P_1\colon \mbox{type}(m(p)) = m_S(\mbox{type}(p)) := m_S(s_1)\times\cdots\times m_S(s_{ar(p)}) \longrightarrow m_S(s)$ for predicates $p\in P_1$ with type$(p)= s_1\times\cdots\times s_{ar(p)}$. Then the signature morphism $m$ can also be extended to a morphism $m^\ast\colon L(\Sigma_1,P_1,X)\longrightarrow L(\Sigma_2,P_2,X')$ between formulas. Such an extension is called a <i>translation</i> as in the case of terms.  
 
  
 
===References===
 
===References===
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|valign="top"|{{Ref|M89}}||valign="top"| B. Möller: "Algorithmische Sprachen und Methodik des Programmierens I", lecture notes, Technical University Munich 1989
 
|valign="top"|{{Ref|M89}}||valign="top"| B. Möller: "Algorithmische Sprachen und Methodik des Programmierens I", lecture notes, Technical University Munich 1989
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|valign="top"|{{Ref|ST99}}||valign="top"| D. Sannella, A. Tarlecki, \"Algebraic Preliminaries \", in Egidio Astesiano, Hans-Joerg Kreowski, Bernd Krieg-Brueckner, "Algebraic Foundations of System Specification", Springer 1999
 
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|valign="top"|{{Ref|W90}}||valign="top"| M. Wirsing: "Algebraic Specification", in J. van Leeuwen: "Handbook of Theoretical Computer Science", Elsevier 1990
 
|valign="top"|{{Ref|W90}}||valign="top"| M. Wirsing: "Algebraic Specification", in J. van Leeuwen: "Handbook of Theoretical Computer Science", Elsevier 1990
 
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Revision as of 11:14, 1 February 2013

$\Sigma$-Algebra

2020 Mathematics Subject Classification: Primary: 68P05 [MSN][ZBL]


$\Sigma$-Algebras are the semantical counterpart to the signatures, which are pure syntactical objects. In order to give the function symbols $f\in F$ of a signature $\Sigma=(S,F)$ a meaning, a (total) $\Sigma$-algebra provides an object with the same structure as $\Sigma$ but consisting of concrete elements and concrete functions operating on these elements. The elements and functions of a $\Sigma$-algebra are the counterparts to the sorts and function symbols of the signature $\Sigma$.

Definition of $\Sigma$-Algebras

Formally, a $\Sigma$-algebra $A=((s^A)_{s\in S},(f^A)_{f\in F})$ consists of a family $(s^A)_{s\in S}$ of carrier sets $s^A$ corresponding to the sorts $s\in S$ and a family $(f^A)_{f\in F}$ of functions on these carrier sets corresponding to the function symbols $f\in F$. The compatibility requirement is that for a function symbol $f$ of type$(f)= s_1\times\cdots\times s_n \longrightarrow s$, the function $f^A$ must have the form $f^A\colon s_1^A\times\cdots\times s_n^A \longrightarrow s^A$.

Category of $\Sigma$-Algebras

Let $\Sigma=(S,F)$ be a signature and let $A,B$ be $\Sigma$-algebras. A $\Sigma$-algebra-morphism $m\colon A\longrightarrow B$ is a family $(m_s\colon s^A \longrightarrow s^B)_{s\in S}$ of mappings between the carrier sets of $A,B$ fulfilling the following compatibility properties

  • $m_s(f^A)= f^B$ for $f\in F$ with ar$(f)=0$ and $\mathrm{type}(f)=\,\, \rightarrow s\}$
  • $m_s(f^A(a_1,\ldots,a_n))=f^B(m_{s_1}(a_1),\ldots,m_{s_n}(a_n))$ for $f\in F$ with $\mathrm{type}(f)= s_1\times\cdots\times s_n \longrightarrow s$ and for $a_i\in s_i^A$.

The class of $\Sigma$-algebras together with the $\Sigma$-algebra-morphisms forms a category [W90].

The option to use partially defined functions complicates the situation considerably and leads to refined versions of the definition. Though this does not belong to the scope of this entry, a short remarks seems to be appropriate. For example it is possible under this generalization that $m_s(f^A(a_1,\ldots,a_n))$ is undefined though $f^A(a_1,\ldots,a_n)$ is defined. This can be caused by an undefinedness of a term $m_{s_j}(a_j)$ or of $f^B(m_{s_1}(a_1),\ldots,m_{s_n}(a_n))$ [M89].

$\Sigma$-Subalgebras

One can easily imagine that typically many different $\Sigma$-algebras exist for the same signature $\Sigma$. This holds even in the case of algebraic specifications, which can restrict the set of admissable $\Sigma$-algebras by additional axioms. In effect, this means that an abstract signature $\Sigma$ can be 'implemented' by concrete $\Sigma$-algebras with different semantics. This leads to an interest in the relationships between these $\Sigma$-algebras. One method to discuss the relationships is to use $\Sigma$-algebra-morphisms, another is the notion of $\Sigma$-subalgebras.

$\Sigma$-Subalgebras of $\Sigma$-algebras are defined in the usual way. A $\Sigma$-Algebra $A$ is called a $\Sigma$-subalgebra of a $\Sigma$-Algebra $B$, if $\forall s\in S\colon s^A\subseteq s^B$ and if $f^A(a_1,\ldots,a_n) = f^B(a_1,\ldots,a_n)$ for $f\in F$ with $\mathrm{type}(f)= s_1\times\cdots\times s_n \longrightarrow s$ and for $a_i\in s_i^A$. The subalgebra-property is written as $A\subseteq B$.

Another way to characterize the subalgebra-property is the existence of a $\Sigma$-algebra-morphism $m\colon A\longrightarrow B$, which is the identity on each carrier set of $A$, i.e. $m=(\mathrm{id}_{s^A})_{s\in S}$.

(Carriers of) $\Sigma$-Subalgebras are closed under intersection. For a family $((s^{A_i})_{s\in S},(f^{A_i})_{f\in F})_{i\in I}$ of $\Sigma$-subalgebras $A_i$, their intersection $A=((s^{A})_{s\in S}, (f^{A})_{f\in F})$ is a $\Sigma$-subalgebra given by carrier sets $s^A:= \bigcap\limits_{i\in I} s^{A_i}$ for all $s\in S$ and functions $f^A:= f^{A_k}|_{s^A}$ for all $f\in F$ for an arbitrarily chosen $k\in I$. The declaration $f^A$ is well-defined, because according to the definition of a $\Sigma$-subalgebra the functions $f^{A_i}$ must behave in the same way on $s^A$.

The closedness of the set of $\Sigma$-subalgebras under intersections has an important consequence. For a $\Sigma$-algebra $A=((s^A)_{s\in S}, (f^A)_{f\in F})$ and an $S$-sorted set $X=(\bar s)_{s\in S}$ with $\bar s\subseteq s^A$ for all $s\in S$, it assures the existence of a smallest $\Sigma$-subalgebra $A'\subseteq A$ of $A$ containing $X$, i.e. $\bar s\subseteq s^{A'}$. The $\Sigma$-algebra $A'$ is called the $\Sigma$-subalgebra of $A$ generated by $X$ [ST99].

The subalgebra-property is compatible with $\Sigma$-algebra-morphisms. Let $A,B$ be $\Sigma$-algebras and let $m=(m_s)_{s\in S}\colon A \longrightarrow B$ be a $\Sigma$-algebra-morphism. For a $\Sigma$-subalgebra $A'\subseteq A$ of $A$, its image $m(A')\subseteq B$ is a $\Sigma$-subalgebra of $B$. The expression $B':=m(A')$ is defined in the obvious way, i.e. for $A'=((s^{A'})_{s\in S},(f^{A'})_{f\in F})$ the $\Sigma$-subalgebra $B'=((s^{B'})_{s\in S},(f^{B'})_{f\in F})$ is given by $s^{B'}:=\{m_s(a)|a\in s^{A'}\}$ for all $s\in S$ and $f_{B'}(m_{s_1}(a_1),\ldots,m_{s_n}(a_n)) = m_s(f_{A'}(a_1,\ldots,a_n))$ for all function symbols $f \colon s_1 \times \ldots\times s_n \longrightarrow s$ with $f\in F$ and $a_i \in s_i^{A'}$. The coimage of a $\Sigma$-subalgebra $B'\subseteq B$ of $B$ under $m$ is a $\Sigma$-subalgebra $m^{-1}(B')\subseteq A$ of $A$. The expression $m^{-1}(B')$ can be defined analogously to $m(A')$ and is omitted here [ST99].

References

[EM85] H. Ehrig, B. Mahr: "Fundamentals of Algebraic Specifications", Volume 1, Springer 1985
[EM90] H. Ehrig, B. Mahr: "Fundamentals of Algebraic Specifications", Volume 2, Springer 1990
[M89] B. Möller: "Algorithmische Sprachen und Methodik des Programmierens I", lecture notes, Technical University Munich 1989
[ST99] D. Sannella, A. Tarlecki, \"Algebraic Preliminaries \", in Egidio Astesiano, Hans-Joerg Kreowski, Bernd Krieg-Brueckner, "Algebraic Foundations of System Specification", Springer 1999
[W90] M. Wirsing: "Algebraic Specification", in J. van Leeuwen: "Handbook of Theoretical Computer Science", Elsevier 1990
How to Cite This Entry:
Joachim Draeger/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Joachim_Draeger/sandbox&oldid=29361