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Distributive implication groupoids

100%
Chajda I., Halaš R.
Open Mathematics
|
2007
|
tom 5
|
nr 3
484-492
EN
We introduce a concept of implication groupoid which is an essential generalization of the implication reduct of intuitionistic logic, i.e. a Hilbert algebra. We prove several connections among ideals, deductive systems and congruence kernels which even coincide whenever our implication groupoid is distributive.
2
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On some congruences of power algebras

68%
Pilitowska A., Zamojska-Dzienio A.
Open Mathematics
|
2012
|
tom 10
|
nr 3
987-1003
EN
In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class of non-trivial subvarieties of the variety of all semilattice ordered modes (modals).
3
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The subalgebra lattice of a finite algebra

60%
Pióro K.
Open Mathematics
|
2014
|
tom 12
|
nr 7
1052-1108
EN
The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.
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