Unary Algebra exists where all axioms of Huntington but one are true

[rsdmath]

I need to prove that there is unary algebra(with 1 element only), that all axioms of Huntington, except the the one that requires at least two elements are true for it.

I tried to think about it, but I don't see how the axioms of A * A' = 0, and A + A' = 1 would be valid for unary algebra with only {1}, or {0} as it's element.
In addition, how do you verify the statements x+ 0 = x and x * 1 = x are true, as in unary algebra we have either 0 or 1, but not both.

Any idea?

Thank you.

 

[Ishuda]

I need to prove that there is unary algebra(with 1 element only), that all axioms of Huntington, except the the one that requires at least two elements are true for it.

I tried to think about it, but I don't see how the axioms of A * A' = 0, and A + A' = 1 would be valid for unary algebra with only {1}, or {0} as it's element.
In addition, how do you verify the statements x+ 0 = x and x * 1 = x are true, as in unary algebra we have either 0 or 1, but not both.

Any idea?

Thank you.

What is an unary Algebra? Is it an Algebra with one element or one with one operation defined on the set? Or do you mean a unary Algebra with one element?

 

[rsdmath]

I mean unary algebra with only one element in the set and +,- as operators.

 

[stapel]

I mean unary algebra with only one element in the set and +,- as operators.

According to online resources, a "unary algebra" is defined as "A universal algebra \(\displaystyle \, \Bigg \langle\, A,\, \left\{ \, f_i\, :\, i\, \in\, I\, \right\} \, \Bigg \rangle\, \) with a family \(\displaystyle \, \left\{\, f_i\, :\, i\, \in\, I\, \right\}\,\) of unary operations \(\displaystyle \, f_i\, :\, A\, \rightarrow\, A.\)"

How does the above relate to what you're working with? Thank you.

 

[Ishuda]

I mean unary algebra with only one element in the set and +,- as operators.

A unary algebra only has one operation defined. I guess by - you mean 'the inverse of +', i.e. if, for a belonging to the unary algebra there exists an element b belonging to the unary algebra such that
a+b=the unit element=0
we call b the inverse of a and write
a-b = 0. Note that not elements of the algebra have to have an inverse.

BTW, to demonstrate existence all you have to do is give one example. Have you considered the set {0}?

 

[Limax]

I need to prove that there is unary algebra(with 1 element only), that all axioms of Huntington, except the the one that requires at least two elements are true for it.

Are you thinking of a Robbins algebra? As I understand it, this is a (nonempty) set \(\displaystyle A\) together with two binary operations \(\displaystyle \vee\) and \(\displaystyle \wedge\) and a unary operation \(\displaystyle \neg\) (a mapping from the set to itself) such that the four axioms on the wiki page hold. Now if \(\displaystyle A=\{a\}\) is just a singleton, then obviously \(\displaystyle \neg a=a\) (there is no other possibility). Hence, just take \(\displaystyle A=\{0\}\) and \(\displaystyle \vee=+,\ \wedge=\times\).