Showing papers on "Idempotence published in 1983"

A Deterministic Algorithm for Factorizing Polynomials of Fq [X]

Abstract: The general problem under consideration is actually solved for every semi-simple, commutative algebra A of finite dimension over a finite field F q . A probabilistic algorithm for decomposing an idempotent u of A into a sum of primitive idempotents was introduced in [1] and [2]. Here the probabilistic algorithm gives place to a deterministic one which, in particular, allows the factorization of polynomials of F q , [ X ].

Idempotent distributive semirings II

Abstract: In [2] it is shown that every idempotent distributive semiring is the Plonka sum of a semilattice ordered system of idempotent distributive semirings which satisfy the generalized absorption law x+xyx+x=x. We shall show that an idempotent distributive semiring which satisfies the above absorption law must be a subdirect product of a distributive lattice and a semiring which satisfies the additional identity xyx+x+xyx=xyx. Using this, we construct the lattice of all equational classes of idempotent distributive semirings for which the two reducts are normal bands.

Varieties with modular subalgebra lattices

Abstract: A classification is begun of varieties of algebras with the property that each algebra in the variety has a modular lattice of subalgebras. This turns out to be a very restrictive condition. Such a variety is hamiltonian. If the algebras in it are idempotent, then it is a variety of sets. A variety is subalgebra-modular if and only if it is hamiltonian and satisfies certain conditions on the terms in its three generator free algebras.

Semihereditary and fully idempotent fpf rings

Abstract: (1983). Semihereditary and fully idempotent fpf rings. Communications in Algebra: Vol. 11, No. 3, pp. 227-242.

Finitely Boolean representable varieties

Abstract: This paper gives a short, elementary proof of a result of Burris and McKenzie [2] stating that each variety Boolean representable by a finite set of finite algebras is the join of an abelian and a discriminator variety. An example showing that the Boolean product operator Ia is not idempotent is included as well.

Full embeddability into categories of generalized algebras

Abstract: Representability of monoids by endomorphisms of algebras from a given class C is often decided by a category-theoretical approach: a full embedding of a binding category B into C is constructed. Since every category of algebras is isomorphic to a full subcategory of any binding category and each small category occurs as a full subcategory of B, every monoid is then isomorphic to the endomorphism monoid of an algebra in C. Furthermore, by the Hedrlfn-KuSera Theorem [19], under the set-theoretical axiom (M), saying that there is only a set of strongly measurable cardinals, any concrete category can be fully embedded into any binding category B. There are numerous binding varieties of algebras. Thus, for instance, Hedrl in and Pultr [7] proved that a category of algebras of a given type zl is binding ff and only if the sum of Zl is bigger than 1. The list includes semigroups [8], commutative rings with unit [3, 4], lattices with (0, 1)-homomorphisms [6], and a locally finite lattice variety [2], to name a few. This leads to a problem of characterization of binding varieties. Two non-structural characterizations are given in Rosicl@ [23] where it is shown that under (M) every concrete complete, cocomplete, locally and colocally small category is binding if and only if a certain two-object category can be fully embedded into it, and in Sichler [26], where the representability of a finite category similarly decides whether a variety of unary algebras is binding. A structural theorem appears in Pultr and Sichler [21] where all varieties of idempotent unary algebras with two operations which are binding are described by identities. The aim of this paper is a characterization of binding (and almost-binding) varieties among those definable by identities in basic operations only. We prove: let V be a variety of algebras of type A given by identities containing only basic operations. Then V is almost-binding if and only if the sum of arities of operations in A is bigger than 1 and it is binding if and only if it is

QUASI Transitive Algebras

Abstract: : Quasitransitive algebras are (extensions of) both Kleene and Stone algebras supplied with: two additional, binary, operations: an equivalential operation, and the 'overmeet', (carrot) -which differs from ordinary meet in lacking idempotence, and in the join's not being distributive into it-; and with an additional unary operation, n, which carries every entity into its lower threshold. The filter of dense elements is considered the truth filter, in virtue of the endorsement principle: what is, to some extent or other however small, true is true. The fuzzy sentential calculus Ap corresponding to those algebras is shown to be both sound and complete.