1 Introduction
2 Preliminaries
2.1 Generalizations of BCI algebras
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An aRM** algebra is an aRM algebra satisfying (**).
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A BCH** algebra is a BCH algebra satisfying (**).
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An *aRM** algebra is an aRM** algebra satisfying (*).
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An aRM**(D1) algebra (resp. *aRM**(D1) algebra ) is an aRM** algebra (resp. *aRM** algebra) satisfying (D1).
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A BZ algebra is an aRM algebra satisfying (B).
2.2 Generalizations of groups
3 The (strong) p-semisimple property
3.1 p-semisimple and strongly p-semisimple algebras
3.2 Connections between RM algebras, moons and goops
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strongly p-semisimple RM algebras and involutive moons,
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strongly p-semisimple aRM algebras and weakly goops,
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strongly p-semisimple aRM** algebras and goops.