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4

You state: infinite unions can have no relationship between strings of a language. Can you explain what you mean by this? Meanwhile, every language is the infinite union of the singleton languages that each contain one element of the language. Clearly, not every language is regular.

7

Look at $$\ell=\{a^p\mid p\text{ is prime}\}.$$ This language obtain from infinite union of $$\bigcup_{i\geq 2, i\text{ is prime}}^{\infty}L_i$$ Where each $L_i=\{a^i\mid i\text{ is prime}\}$ that have one word. Another example is $\{a^nb^n\mid n\in\mathbb{N}\}$ That isn't regular and we can describe it by infinite union of regular languages \bigcup_{i\geq ...

2

The symbol $(\bot)$ usually represents the least element of a lattice. The least element often goes at the bottom part of a Hasse diagram. That is maybe the reason you're looking for. For example, in a Hasse diagram containing propositions with the logical consequence relation, one usually put falsehood, at the bottom of the diagram, mostly because from a ...

0

Meta-variable explanation by example Consider the simply typed lambda calculus (STLC) with base types Int for integers and Bool for booleans. In this calculus, we can have an identity function for integers $(\lambda x:\mathrm{Int}.x)$ and another one for booleans $(\lambda x:\mathrm{Bool}.x)$. More generally, For every type $A$ of STLC, there is an identity ...

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