Let G be a complex simple algebraic group with Lie algebra g. Fix a Borel subalgebra b. An ideal of b is called ad-nilpotent, if it is contained in [b, b]. The goal of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals in the case, where g has roots of different length. Let Ad denote the set of all ad-nilpotent ideals of b. Any c ∈ Ad is completely determined...

in this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. it is proved that an abelian group of finite exponent is strongly coatomic.

This study aims to determine the concentration of SLS and NaCl which can produce best formula aloe vera extract liquid soap (Aloe L.). The method used in this is Simplex Lattice Design method. Eight formulas were made with different concentrations NaCl, then parameters viscosity, specific gravity, foam resistance, pH determined. data obtained was tested one sample t-test see difference between ...

the aim of this paper is to extend results established by h. onoand t. kowalski regarding directly indecomposable commutative residuatedlattices to the non-commutative case. the main theorem states that a residuatedlattice a is directly indecomposable if and only if its boolean center b(a)is {0, 1}. we also prove that any linearly ordered residuated lattice and anylocal residuated lattice are d...