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On ideal theory of hoops

Mona Aaly Kologani, Rajab Ali Borzooei (2020)

Mathematica Bohemica

In this paper, we define and characterize the notions of (implicative, maximal, prime) ideals in hoops. Then we investigate the relation between them and prove that every maximal implicative ideal of a ∨ -hoop with double negation property is a prime one. Also, we define a congruence relation on hoops by ideals and study the quotient that is made by it. This notion helps us to show that an ideal is maximal if and only if the quotient hoop is a simple MV-algebra. Also, we investigate the relationship...

On ideals and congruences in BCC-algebras

Wiesław Aleksander Dudek, Xiaohong Zhang (1998)

Czechoslovak Mathematical Journal

We introduce a new concept of ideals in BCC-algebras and describe connections between such ideals and congruences.

On ideals in De Morgan residuated lattices

Liviu-Constantin Holdon (2018)

Kybernetika

In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal...

On idempotent modifications of M V -algebras

Ján Jakubík (2007)

Czechoslovak Mathematical Journal

The notion of idempotent modification of an algebra was introduced by Ježek. He proved that the idempotent modification of a group is subdirectly irreducible. For an M V -algebra 𝒜 we denote by 𝒜 ' , A and ℓ ( 𝒜 ) the idempotent modification, the underlying set or the underlying lattice of 𝒜 , respectively. In the present paper we prove that if 𝒜 is semisimple and ℓ ( 𝒜 ) is a chain, then 𝒜 ' is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras.

On infinite partitions of lines and space

Paul Erdös, Steve Jackson, R. Mauldin (1997)

Fundamenta Mathematicae

Given a partition P:L → ω of the lines in ℝ n , n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, Q : ℝ n → ω , so that each line meets at most m points of its color. Assuming Martin’s Axiom, we show this is the case for m ≥ 3. We reduce the problem for m = 2 to a purely finitary geometry problem. Although we have established a very similar, but somewhat simpler, version of the geometry conjecture, we leave the general problem open. We consider also various generalizations...

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