We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector Lattices. In this regards, some characterizations of local vector Lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-trivial components. Nevertheless, our main purpose is to prove, via what we call the radical functor, that the category of all vector Lattices and lattice homomorphisms is equivalent to the category of local vectors Lattices and unital (i. e., unit preserving) lattice homomorphisms.

The systematic study of planar semimodular Lattices started in 2007 with a series of papers by G. Gr¨, atzer and E. Knapp. These Lattices have connections with group theory and geometry. A planar semimodular lattice L is slim if M3 it is not a sublattice of L. In his 2016 monograph, “, The Congruences of a Finite Lattice, A Proof-by-Picture Approach”, , the second author asked for a characterization of congruence Lattices of slim, planar, semimodular Lattices. In addition to distributivity, both authors have previously found specific properties of these congruence Lattices. In this paper, we present a new property, the Three-pendant Three-crown Property. The proof is based on the first author’, s papers: 2014 (multifork extensions), 2017 (C1-diagrams), and a recent paper (lamps), introducing the tools we need.

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.

In this paper, the notion of mp-residuated lattice, as a subclass of residuated Lattices in which every prime filter contains a unique minimal prime filter, is introduced and investigated. For a residuated lattice A, the notion of ω-filter is introduced and it is shown that Ω (A), the set of ω-filters of A, is a bounded distributive lattice. Also, it is observed that γ (A), the set of coannulets of A, is a sublattice of Ω (A). Then for each prime filter P of A, the notion of the divisor filter D(P) as an important tool in investigating of minimal prime filters of A is introduced and it is proved that a prime filter P is minimal prime if and only if P=D(P). Finally, by the notion of ω-filters, as an extension of divisor filters, a fundamental characterization of mp-residuated Lattices is given and it is shown that a residuated lattice is mp if and only if the set of its ω-filters is a sublattice of the lattice of its filters.

In this paper, we study the separtion axioms T 0, T 1, T 2 and T 5/2 on topological and semitopological residuated Lattices and we show that they are equivalent on topological residuated Lattices. Then we prove that for every infinite cardinal number a, there exists at least one nontrivial Hausdor topological residuated lattice of cardinality a. In the follows, we obtain some conditions on (semi) topological residuated Lattices under which this spaces will convert into regular and normal spaces.Finally by using of regularity and normality, we convert (semi) topological residuated Lattices into metrizable topological residuated Lattices.

In this paper, we study residuated Lattices in order to give new characterizations for dense, regular and Boolean elements in residuated Lattices and investigate special residuated Lattices in order to obtain new characterizations for the directly indecomposable subvariety of Stonean residuated Lattices. Free algebra in varieties of Stonean residuated Lattices is constructed. We introduce in residuated lattice a new type of filter called special filter and investigate its properties. Finally, regular filter property in residuated Lattices is introduced and is studied in details.

This paper introduces and investigates the notion of a generalized Stone residuated lattice. It is observed that a residuated lattice is generalized Stone if and only if it is quasicomplemented and normal. Also, it is proved that a , nite residuated lattice is generalized Stone if and only if it is normal. A characterization for generalized Stone residuated Lattices is given by means of the new notion of α,-, filters. Finally, it is shown that each non-unit element of a directly indecomposable generalized Stone residuated lattice is a dense element.