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Characterizing cryptogroups with a finite number of $${\mathcal{H}}$$ -classes in each $${\mathcal{D}}$$ -class by their subsemigroups
Authors:Mario Petrich
Institution:(1) 21420 Bol, Brač, Croatia
Abstract:We construct a family $${\mathcal{F}}$$ of completely regular semigroups with the property that each completely regular semigroup S with a finite number of $${\mathcal{H}}$$ -classes in each $${\mathcal{D}}$$ -class is non-cryptic if and only if S contains an isomorphic image of a member of $${\mathcal{F}}$$ . Each member F of $${\mathcal{F}}$$ is an ideal extension of a Rees matrix semigroup J by a cyclic group B with a zero adjoined and the identity of B is the identity of F. Here $$J={\mathcal{M}}\left( I,G,\Lambda;P\right) $$ with I and Λ finite, G is given by generators and relations, and P is given explicitly. Within completely regular semigroups, the cryptic property is equivalent to $${\mathcal{N}=\mathcal{S}},$$ where $${\mathcal{N}}$$ is the natural partial order and a $${\mathcal{S}} b$$ if and only if a 2 = ab = ba. Hence the above result can be formulated in terms of $${\mathcal{N}}$$ and $${\mathcal{S}}$$ .
Keywords:Completely regular semigroup  Cryptic  Cryptogroup  Natural partial order  Rees matrix semigroup  Ideal extension
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