Some Special Artex Spaces Over Bi-monoids

Abstract

We introduce a Special Artex Space over a bi-monoid namely Distributive Artex space over a bi-monoid. We give some examples of Distributive Artex space over bi-monoids. We prove the Cartesian product of any two Distributive Artex Spaces over a bi-monoid is a Distributive Artex Space over the bi-monoid. Also we prove the Cartesian product of a finite number of Distributive Artex Spaces over a bi-monoid is a Distributive Artex Space over the bi-monoid. We prove under the Artex space homomorphism f : A ? B, the homomorphic image of a SubArtex Space of an Artex space A over a bi-monoid is a SubArtex space of B. We prove the homomorphic image of a Distributive Artex Space over a bi-monoid is a Distributive Artex Space over the bi-monoid. We  prove a SubArtex space of a Distributive Artex space over a bi-monoid  is a Distributive Artex space over the bi-mpnoid. We solve three problems on Bounded Artex spaces over bi-monoids. 1.A SubArtex space of a Lower Bounded Artex space  over a bi-monoid need not be a Lower Bounded Artex space over the bi-monoid. 2. A SubArtex space of an Upper Bounded Artex space  over a bi-monoid need not be an Upper Bounded Artex space over the bi-monoid and 3. A SubArtex space of a Bounded Artex space  over a bi-monoid need not be a Bounded Artex space over the bi-monoid.

Keywords : Distributive Artex space, Bounded Artex space, Homomorphic image