Is there a generalization of universal algebra in which inequalities are permitted?
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In ordinary universal algebra, we consider only equational axioms like $$x + y = y + x, qquad x + -x = 0.$$
There's a generalization of universal algebra in which quasi-equations are permitted, which are basically implications between finite conjunctions of equations. For example, the notion of a cancellative monoid can be axiomatized by quasi-equations; the left-cancellation law is
$$ax = ay rightarrow x = y$$
It seems reasonable to consider a variant on this in which inequalities are also fair game. For example, to axiomatize integral domains, we might use an appropriate cancellation law:
$$a neq 0 wedge ax = ay rightarrow x = y.$$
Alternatively, we could use a version of the null-factor law: $$ab = 0 wedge a neq 0 rightarrow b = 0.$$
I envisage that the morphisms between models would be taken as injective homomorphisms. Such categories won't usually have products or a terminal object, though they'll often have an initial object and/or coproducts.
Here's an example of how such categories might be useful.
Definition. A non-degenerate Peano structure consists of a triple $(X,S,0)$ where $X$ is a set, $S :X rightarrow X$ is an injective function, $0 in X$ is an element, and $forall x in X(S(x) neq 0)$ is assumed to hold.
The informal statement that every non-degenerate Peano structure contains a natural copy of $mathbb{N}$ can be formally stated as "$mathbb{N}$ is the initial non-degenerate Peano structure."
Question. Can any interesting results be proved for category of models of "quasi-equations with negation", where the morphisms are taken to be injective homomorphisms?
abstract-algebra logic category-theory universal-algebra
asked Nov 30 '18 at 11:49
goblingoblin
36.9k1159193
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add a comment |
$begingroup$
In ordinary universal algebra, we consider only equational axioms like $$x + y = y + x, qquad x + -x = 0.$$
There's a generalization of universal algebra in which quasi-equations are permitted, which are basically implications between finite conjunctions of equations. For example, the notion of a cancellative monoid can be axiomatized by quasi-equations; the left-cancellation law is
$$ax = ay rightarrow x = y$$
It seems reasonable to consider a variant on this in which inequalities are also fair game. For example, to axiomatize integral domains, we might use an appropriate cancellation law:
$$a neq 0 wedge ax = ay rightarrow x = y.$$
Alternatively, we could use a version of the null-factor law: $$ab = 0 wedge a neq 0 rightarrow b = 0.$$
I envisage that the morphisms between models would be taken as injective homomorphisms. Such categories won't usually have products or a terminal object, though they'll often have an initial object and/or coproducts.
Here's an example of how such categories might be useful.
Definition. A non-degenerate Peano structure consists of a triple $(X,S,0)$ where $X$ is a set, $S :X rightarrow X$ is an injective function, $0 in X$ is an element, and $forall x in X(S(x) neq 0)$ is assumed to hold.
The informal statement that every non-degenerate Peano structure contains a natural copy of $mathbb{N}$ can be formally stated as "$mathbb{N}$ is the initial non-degenerate Peano structure."
Question. Can any interesting results be proved for category of models of "quasi-equations with negation", where the morphisms are taken to be injective homomorphisms?
abstract-algebra logic category-theory universal-algebra
asked Nov 30 '18 at 11:49
goblingoblin
36.9k1159193
$endgroup$
1
$begingroup$
I suspect that these categories generally won't be nice, since fields are axiomatizable by quasi-equations with negation. (Commutative unital ring axioms plus a unary function $xmapsto x^{-1}$ satisfying $xne 0 to x(x^{-1})=1$)
$endgroup$
– jgon
Dec 1 '18 at 3:21
1
$begingroup$
@jgon, yes, that's a good point. However, I'd probably call that a "pointed field", since it consists of a field together with a choice of value of $x^{-1}$. Following meadow theory, we should probably define $0^{-1} = 0$. This makes no sense topologically, but algebraically it's the best option. And that way, we're not dealing with "pointed fields", but rather with bona-fide fields.
$endgroup$
– goblin
Dec 1 '18 at 12:12
$begingroup$
Oh yes, good point, sorry, that was my intent, but words to express that didn't end up on the page.
$endgroup$
– jgon
Dec 1 '18 at 13:15
$begingroup$
Allegories are a form of categories defined by inclusions rather than equations, moreover their variations are axiomatised by Horn clauses; so that might be something to look into for insight --see the standard text Allegories, Categories ;-)
$endgroup$
– Musa Al-hassy
Dec 5 '18 at 20:41
add a comment |
$begingroup$
In ordinary universal algebra, we consider only equational axioms like $$x + y = y + x, qquad x + -x = 0.$$
There's a generalization of universal algebra in which quasi-equations are permitted, which are basically implications between finite conjunctions of equations. For example, the notion of a cancellative monoid can be axiomatized by quasi-equations; the left-cancellation law is
$$ax = ay rightarrow x = y$$
It seems reasonable to consider a variant on this in which inequalities are also fair game. For example, to axiomatize integral domains, we might use an appropriate cancellation law:
$$a neq 0 wedge ax = ay rightarrow x = y.$$
Alternatively, we could use a version of the null-factor law: $$ab = 0 wedge a neq 0 rightarrow b = 0.$$
I envisage that the morphisms between models would be taken as injective homomorphisms. Such categories won't usually have products or a terminal object, though they'll often have an initial object and/or coproducts.
Here's an example of how such categories might be useful.
Definition. A non-degenerate Peano structure consists of a triple $(X,S,0)$ where $X$ is a set, $S :X rightarrow X$ is an injective function, $0 in X$ is an element, and $forall x in X(S(x) neq 0)$ is assumed to hold.
The informal statement that every non-degenerate Peano structure contains a natural copy of $mathbb{N}$ can be formally stated as "$mathbb{N}$ is the initial non-degenerate Peano structure."
Question. Can any interesting results be proved for category of models of "quasi-equations with negation", where the morphisms are taken to be injective homomorphisms?
abstract-algebra logic category-theory universal-algebra
asked Nov 30 '18 at 11:49
goblingoblin
36.9k1159193
$endgroup$
In ordinary universal algebra, we consider only equational axioms like $$x + y = y + x, qquad x + -x = 0.$$
There's a generalization of universal algebra in which quasi-equations are permitted, which are basically implications between finite conjunctions of equations. For example, the notion of a cancellative monoid can be axiomatized by quasi-equations; the left-cancellation law is
$$ax = ay rightarrow x = y$$
It seems reasonable to consider a variant on this in which inequalities are also fair game. For example, to axiomatize integral domains, we might use an appropriate cancellation law:
$$a neq 0 wedge ax = ay rightarrow x = y.$$
Alternatively, we could use a version of the null-factor law: $$ab = 0 wedge a neq 0 rightarrow b = 0.$$
I envisage that the morphisms between models would be taken as injective homomorphisms. Such categories won't usually have products or a terminal object, though they'll often have an initial object and/or coproducts.
Here's an example of how such categories might be useful.
Definition. A non-degenerate Peano structure consists of a triple $(X,S,0)$ where $X$ is a set, $S :X rightarrow X$ is an injective function, $0 in X$ is an element, and $forall x in X(S(x) neq 0)$ is assumed to hold.
The informal statement that every non-degenerate Peano structure contains a natural copy of $mathbb{N}$ can be formally stated as "$mathbb{N}$ is the initial non-degenerate Peano structure."
Question. Can any interesting results be proved for category of models of "quasi-equations with negation", where the morphisms are taken to be injective homomorphisms?
abstract-algebra logic category-theory universal-algebra
abstract-algebra logic category-theory universal-algebra
asked Nov 30 '18 at 11:49
goblingoblin
36.9k1159193
asked Nov 30 '18 at 11:49
goblingoblin
36.9k1159193
edited Dec 1 '18 at 12:12
goblin
asked Nov 30 '18 at 11:49
goblingoblin
36.9k1159193
asked Nov 30 '18 at 11:49
goblingoblin
36.9k1159193
asked Nov 30 '18 at 11:49
goblingoblin
36.9k1159193
36.9k1159193
1
$begingroup$
I suspect that these categories generally won't be nice, since fields are axiomatizable by quasi-equations with negation. (Commutative unital ring axioms plus a unary function $xmapsto x^{-1}$ satisfying $xne 0 to x(x^{-1})=1$)
$endgroup$
– jgon
Dec 1 '18 at 3:21
1
$begingroup$
@jgon, yes, that's a good point. However, I'd probably call that a "pointed field", since it consists of a field together with a choice of value of $x^{-1}$. Following meadow theory, we should probably define $0^{-1} = 0$. This makes no sense topologically, but algebraically it's the best option. And that way, we're not dealing with "pointed fields", but rather with bona-fide fields.
$endgroup$
– goblin
Dec 1 '18 at 12:12
$begingroup$
Oh yes, good point, sorry, that was my intent, but words to express that didn't end up on the page.
$endgroup$
– jgon
Dec 1 '18 at 13:15
$begingroup$
Allegories are a form of categories defined by inclusions rather than equations, moreover their variations are axiomatised by Horn clauses; so that might be something to look into for insight --see the standard text Allegories, Categories ;-)
$endgroup$
– Musa Al-hassy
Dec 5 '18 at 20:41
add a comment |
1
$begingroup$
I suspect that these categories generally won't be nice, since fields are axiomatizable by quasi-equations with negation. (Commutative unital ring axioms plus a unary function $xmapsto x^{-1}$ satisfying $xne 0 to x(x^{-1})=1$)
$endgroup$
– jgon
Dec 1 '18 at 3:21
1
$begingroup$
@jgon, yes, that's a good point. However, I'd probably call that a "pointed field", since it consists of a field together with a choice of value of $x^{-1}$. Following meadow theory, we should probably define $0^{-1} = 0$. This makes no sense topologically, but algebraically it's the best option. And that way, we're not dealing with "pointed fields", but rather with bona-fide fields.
$endgroup$
– goblin
Dec 1 '18 at 12:12
$begingroup$
Oh yes, good point, sorry, that was my intent, but words to express that didn't end up on the page.
$endgroup$
– jgon
Dec 1 '18 at 13:15
$begingroup$
Allegories are a form of categories defined by inclusions rather than equations, moreover their variations are axiomatised by Horn clauses; so that might be something to look into for insight --see the standard text Allegories, Categories ;-)
$endgroup$
– Musa Al-hassy
Dec 5 '18 at 20:41
1
1
$begingroup$
I suspect that these categories generally won't be nice, since fields are axiomatizable by quasi-equations with negation. (Commutative unital ring axioms plus a unary function $xmapsto x^{-1}$ satisfying $xne 0 to x(x^{-1})=1$)
$endgroup$
– jgon
Dec 1 '18 at 3:21
$begingroup$
I suspect that these categories generally won't be nice, since fields are axiomatizable by quasi-equations with negation. (Commutative unital ring axioms plus a unary function $xmapsto x^{-1}$ satisfying $xne 0 to x(x^{-1})=1$)
$endgroup$
– jgon
Dec 1 '18 at 3:21
1
1
$begingroup$
@jgon, yes, that's a good point. However, I'd probably call that a "pointed field", since it consists of a field together with a choice of value of $x^{-1}$. Following meadow theory, we should probably define $0^{-1} = 0$. This makes no sense topologically, but algebraically it's the best option. And that way, we're not dealing with "pointed fields", but rather with bona-fide fields.
$endgroup$
– goblin
Dec 1 '18 at 12:12
$begingroup$
@jgon, yes, that's a good point. However, I'd probably call that a "pointed field", since it consists of a field together with a choice of value of $x^{-1}$. Following meadow theory, we should probably define $0^{-1} = 0$. This makes no sense topologically, but algebraically it's the best option. And that way, we're not dealing with "pointed fields", but rather with bona-fide fields.
$endgroup$
– goblin
Dec 1 '18 at 12:12
$begingroup$
Oh yes, good point, sorry, that was my intent, but words to express that didn't end up on the page.
$endgroup$
– jgon
Dec 1 '18 at 13:15
$begingroup$
Oh yes, good point, sorry, that was my intent, but words to express that didn't end up on the page.
$endgroup$
– jgon
Dec 1 '18 at 13:15
$begingroup$
Allegories are a form of categories defined by inclusions rather than equations, moreover their variations are axiomatised by Horn clauses; so that might be something to look into for insight --see the standard text Allegories, Categories ;-)
$endgroup$
– Musa Al-hassy
Dec 5 '18 at 20:41
$begingroup$
Allegories are a form of categories defined by inclusions rather than equations, moreover their variations are axiomatised by Horn clauses; so that might be something to look into for insight --see the standard text Allegories, Categories ;-)
$endgroup$
– Musa Al-hassy
Dec 5 '18 at 20:41
add a comment |
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I suspect that these categories generally won't be nice, since fields are axiomatizable by quasi-equations with negation. (Commutative unital ring axioms plus a unary function $xmapsto x^{-1}$ satisfying $xne 0 to x(x^{-1})=1$)
$endgroup$
– jgon
Dec 1 '18 at 3:21
1
$begingroup$
@jgon, yes, that's a good point. However, I'd probably call that a "pointed field", since it consists of a field together with a choice of value of $x^{-1}$. Following meadow theory, we should probably define $0^{-1} = 0$. This makes no sense topologically, but algebraically it's the best option. And that way, we're not dealing with "pointed fields", but rather with bona-fide fields.
$endgroup$
– goblin
Dec 1 '18 at 12:12
$begingroup$
Oh yes, good point, sorry, that was my intent, but words to express that didn't end up on the page.
$endgroup$
– jgon
Dec 1 '18 at 13:15
$begingroup$
Allegories are a form of categories defined by inclusions rather than equations, moreover their variations are axiomatised by Horn clauses; so that might be something to look into for insight --see the standard text Allegories, Categories ;-)
$endgroup$
– Musa Al-hassy
Dec 5 '18 at 20:41