Let S be a locally compact foundation semigroup with identity and ( ) be its semigroup algebra. Let X be a weak*-closed left translation invariant subspace of ( ) In this paper, we prove that X is invariantly Complemented in ( ) if and only if the left ideal of ( ) has a bounded approximate identity. We also prove that a foundation semigroup with identity S is left amenable if and only if every Complemented weak*-closed left translation invariant subspace of ( ( )) is invariantly Complemented in ( ( )).

The notion of hemiComplemented lattices is introduced and some of the properties of these algebras are studied. Some characterization theorems of hemiComplemented lattices are derived with the help of minimal prime D-filters, ideals, and congruences. The notion of D-Stone lattices is introduced and then derived a set of equivalent conditions for a hemiComplemented lattice to become a D-Stone lattice. Hemicom-Complemented lattices and D-Stone lattices are characterized in topological terms.

The notion of hemiComplemented lattices is introduced and some of the properties of these algebras are studied. Some characterization theorems of hemiComplemented lattices are derived with the help of minimal prime D-filters, ideals, and congruences. The notion of D-Stone lattices is introduced and then derived a set of equivalent conditions for a hemiComplemented lattice to become a D-Stone lattice. Hemicom-Complemented lattices and D-Stone lattices are characterized in topological terms.

Assume that F denotes a specific space in the class Fα ; p con-structed by H. Khodabakhshian [2] as a class of separable Banach function spaces similar to the well-known James function spaces. In this note, we prove that lp(α ) is isomorphic to a Complemented subspace of Fα ; p, and that Fα ; 2 is a closed subspace of the Waterman-Shiba space α BV 2.

Let Pq(n) be the set of all subspaces in the vector space F n q. There is a subspace distance dS(U, V ) between any two subspaces U and V. A subspace code is also a subset of Pq(n). It is known that dS(U, V ) ≥ dH(ν(πU), ν(πV )), where π ∈ Sn, ν(U) denotes the pivot vector of E(U) and E(U) is the reduced row echelon form of the generator matrix of U. In this paper, we show that if E(U) and E(V ) have at most one non-zero entry in each rows and each columns then the equality holds. Moreover, we introduce the sets GU, V = {π ∈ Sn | dS(U, V ) = dH(ν(πU), ν(πV ))} for any U, V ∈ Pq(n) and examine them in the spaces P2(4), P2(5), P2(6) and P3(4). It is shown that the groups 1, Z2, Z2 × Z2, S3, S4 and 1, Z2, Z2 × Z2, S3, D8, S3 × Z2, S4, S5 appears between these sets in P2(4) and P2(5), respectively. Moreover, the groups 1, Z2, Z2 ×Z2, S3, D8, Z2 ×Z2 ×Z2, S3 ×Z2, D8 ×Z2, S4, S3 × S3, S4 × Z2, (S3 × S3): 2, S5, S6 and 1, Z2, Z2 × Z2, S3, D8, S4 appears between these sets in P2(6) and P3(4), respectively.

We give a general overview of the state-of-the-art in subspace system identification methods. We have restricted ourselves to the most important ideas and developments since the methods appeared in the late eighties. First, the basis of linear subspace identification are summarized. Different algorithms one finds in literature (Such as N4SID, MOESP, CVA) are discussed and put into a unifying framework. Further, a comparison between subspace identification and prediction error methods is made on the basis of computational complexity and precision of methods by applying them to a glass tube manufacturing process.

In this paper, we introduced the concepts of normlet and normal ideal in a pseudo-Complemented almost distributive lattice and studied its properties. We have char-acterized normal ideals and established equivalent conditions for every ideal to become a normal ideal. Also, derived equivalent conditions for the set of all prime normal ideals of a pseudo-Complemented ADL to become a Hausdorff space.