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The property (*) is actually equivalent to a set of quasi-identities: $$(x,y)=(x',y')\rightarrow x=x'$$ $$(x,y)=(x',y')\rightarrow y=y'$$ The converse implication you had ($\leftarrow$) is logically v …

For any vector space $V$ and any function $m:V\to\mathbb N$ you could say you have a vector space $V$ whose underlying set is a multiset with multiplicities given by $m$. This doesn't mean that the ve …

$A'$ is the discrete partition of $A$. That is, we think of it as a partition of $A$ induced by the finest equivalence relation, the identity relation.

Under the Generalized Continuum Hypothesis, $$2^{\aleph_\alpha}=\aleph_{\alpha+1}\quad(\forall\alpha),$$ sets with no structure (so automorphisms are just bijections) is an example. Namely, by Cardi …

Yes, there is now Pavlovic's characterization of Turing computability in terms of the monoidal computer, based on monoidal categories. http://arxiv.org/abs/1208.5205

Notice that by the inference rule $$\frac{M\supset N}{\bigcirc M\supset\bigcirc N}\tag{@}$$ we have $$\frac{\bot\supset N}{\bigcirc \bot\supset\bigcirc N}$$ But $\bot\supset N$ always holds. So either …