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For a left action $S\overset{\lambda}{\curvearrowright}X$ of a cancellativeright amenable monoid $S$ on a discrete Abelian group $X$, we construct its Orelocalization $G\overset{\lambda^*}{\curvearrowright}X^*$, where $G$ is thegroup of left fractions of $S$; analogously, for a right action$K\overset{\rho}\curvearrowleft S$ on a compact space $K$, we construct its Orecolocalization $K^*\overset{\rho^*}{\curvearrowleft} G$. Both constructionspreserve entropy, i.e., for the algebraic entropy $h_{\mathrm{alg}}$ and forthe topological entropy $h_{\mathrm{top}}$ one has$h_{\mathrm{alg}}(\lambda)=h_{\mathrm{alg}}(\lambda^*)$ and$h_{\mathrm{top}}(\rho)=h_{\mathrm{top}}(\rho^*)$, respectively.  Exploiting these constructions and the theory of quasi-tilings, we extend theAddition Theorem for $h_{\mathrm{top}}$, known for right actions of countableamenable groups on compact metrizable groups, to right actions$K\overset{\rho}{\curvearrowleft} S$ of cancellative right amenable monoids $S$(with no restrictions on the cardinality) on arbitrary compact groups $K$.  When the compact group $K$ is Abelian, we prove that $h_{\mathrm{top}}(\rho)$coincides with $h_{\mathrm{alg}}(\hat{\rho})$, where$S\overset{\hat{\rho}}\curvearrowright X$ is the dual left action on thediscrete Pontryagin dual $X=\hat{K}$, that is, a so-called Bridge Theorem. Fromthe Addition Theorem for $h_{\mathrm{top}}$ and the Bridge Theorem, we obtainan Addition Theorem for $h_{\mathrm{alg}}$ for left actions$S\overset{\lambda}\curvearrowright X$ on discrete Abelian groups, so far knownonly under the hypotheses that either $X$ is torsion or $S$ is locallymonotileable.  The proofs substantially use the unified approach towards entropy based onthe entropy of actions of cancellative right amenable monoids on appropriatelydefined normed monoids.

Ore localization of amenable monoid actions and applications towards entropy $-$ addition formulas and the bridge theorem