• IRREDUCIBLE ELEMENTS IN ALMOST LATTICES AND RELATIVELY COMPLEMENTED ALMOST LATTICES

G. NANAJI RAO, HABTAMU TIRUNEH ALEMU*

Abstract


The concepts of atom in an Almost Lattice(AL) L and atomic AL are introduced and proved that every finite AL with 0 is atomic AL. The concepts of meet(join) irreducible elements and meet(join) prime elements are introduced in an AL L and proved that if L is an AL with 0 satisfying minimum(maximum) condition, then every one of its element in L can be represented as the join(meet) of a finite number of join(meet) irreducible elements. Also, a necessary and sufficient condition for an element in an AL,  of two ALs  and  to become join irreducible element is established and proved that every meet(join) prime element is meet(join) irreducible; but, the converse need not be true. The concepts of relatively complemented AL and sectionally complemented AL are introduced and proved that if L is finite and sectionally complemented, then every non zero element of L is a join of finitely many atoms. Further, the concepts of semicomplemented AL and weakly complemented AL are introduced and a necessary and sufficient condition for an AL L with 0 to become a weakly complemented AL is proved. Also, proved that every sectionally complemented AL is semicomplemented.


Keywords


Lattice, Almost Lattice, Almost Lattice with zero, Ascending Chain, Descending Chain, Maximum Condition, Minimum Condition, Atom, Join Irreducible Element, Meet Irreducible Element, Meet Prime Element, Join Prime Element, Inner Element, Maximal Element, R

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