IRREDUCIBLE ELEMENTS IN ALGEBRAIC LATTICES

α-Irreducible and α-Strongly Irreducible Ideals of a ring have been characterized in [2] and [4]. A complete lattice which is generated by compact elements is called an algebraic lattice for the simple reason that every such lattice is isomorphic to the lattice of subalgebras of a suitable universal algebra and vice-versa. In this paper, we characterize the irreducible elements and strongly irreducible elements in an algebraic lattice, which extends the results in [4] to arbitrary algebraic lattices. Also we obtain certain necessary and sufficient conditions, in terms of irreducible elements, for an algebraic lattice to satisfy the complete distributivity.

Modularity* in Lie algebras

A subalgebra U of a Lie algebra L over a field F is called modular* in L if U satisfies the dual of the modular identities in the lattice of subalgebras of L. Our aim is the study of the influence of the modular* identities in the structure of the algebra. First we prove that if the modular* conditions are imposed on an ideal of L then every element of L acts as an scalar on this ideal and if they are imposed on a non-ideal subalgebra U of L then the largest ideal of L contained in U also satisfies the modular* identities. We determine Lie algebras having a subalgebra which satisfies both the modular and modular* identities, provided that either L is solvable or char(F)≠ 2,3. As immediate consequences of this result we obtain that the existence of a co-atom satisfying the modular* identities in the lattice L(L) forces that the lattice L(L) is modular and that the modular* identities on any subalgebra U forces that U is quasi-abelian. In the case when L is supersolvable we obtain that the modular* conditions on any non-ideal of L are enough to guarantee that L(L) is modular. For arbitrary fields and any Lie algebra L, we prove that the modular* conditions on every co-atom of the lattice L(L) guarantee that L(L) is modular.

Covers and complements in the subalgebra lattice of a Boolean algebra

Section 1 addresses the problem of covers in Sub D, the lattice of subalgebras of a Boolean algebra; we describe those BA's in whose subalgebra lattice every element has a cover, and show that every small and separable subalgebra of P(ω) has 2ω covers in SubP(ω). Section 2 is concerned with complements and quasicomplements. As a general result it is shown that Sub D is relatively complemented if and only if D is a finite– cofinite BA. Turning to Sub P(ω), we show that no small and separable D ≤ P(ω) can be a quasicomplement. In the final section, generalisations of packed algebras are discussed, and some properties of these classes are exhibited.

Lattice isomorphisms of modular inverse semigroups

For an inverse semigroup S we will consider the lattice of inverse subsemigroups of S, denoted L(S). A major problem in algebra has been that of finding to what extent an algebra is determined by its lattice of subalgebras. (See, for example, the survey article [9]). By a lattice isomorphism (L-isomorphism, structural isomorphism, or projectivity) of an inverse semigroup S onto another T we shall mean an isomorphism Φ of L(S) onto L(T). A mapping φ from S to T is said to induce Φ if AΦ = Aφ for all A in L(S). We say that S is strongly determined by L(S) if every lattice isomorphism of S onto T is induced by an isomorphism of S onto T.

Lattice isomorphisms of Malcev algebras

SynopsisThe lattice of subalgebras of a Malcev algebra determines to a great extent the structureof the algebra. It is shown that conditions such as nilpotency, solvability or semisimplicity are almost characterised by means of conditions on this lattice. This enables us to study the relationship between Malcev algebras with isomorphic lattices of subalgebras.