3
votes
1answer
83 views

$\mathrm{Pol}_m(\mathbb{A})$ viewed as a relation pp-definable from $\mathbb{A}$

First let me recall some (abbreviated, and possibly simplified to suit my situation) definitions: Let $A$ be a finite set and $\mathbb{A}$ some set of relations on $A$. Let $m, n$ be positive ...
2
votes
2answers
180 views

Smallest Congruence Relation generated by a set

$\newcommand{\cl}{\operatorname{cl}}$ Let $R \subset S \times S$ be a binary relation, the smallest i) reflexive relation containing it is $$ \cl_\mathrm{ref} = R \cup \{ (x,x) : x \in S \} $$ ii) ...
3
votes
3answers
319 views

New kind of identities?

I found a new kind of identities which are half logic and half algebraic while working on a proof of NP-completeness. They are like this: $$ \frac{a+mb}{n+m} < \frac{a}{n} \iff b < ...
2
votes
1answer
631 views

Modus Ponens: implication versus entailment

Would it be inconsistent to write Modus Ponens using only implication, not entailment? $(p \wedge (p \to q)) \to q$ The way I understand is that implication ($ \to$) is an operator that yields a new ...