On the isomorphism problem of concept algebras

Weakly dicomplemented lattices are bounded lattices equipped with two unary operations to encode a negation on {\it concepts}. They have been introduced to capture the equational theory of concept algebras \cite{Wi00}. They generalize Boolean algebras. Concept algebras are concept lattices, thus complete lattices, with a weak negation and a weak opposition. A special case of the representation problem for weakly dicomplemented lattices, posed in \cite{Kw04}, is whether complete {\wdl}s are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem \ref{T:main}). We also provide a new proof of a well known result due to M.H. Stone \cite{St36}, saying that {\em each Boolean algebra is a field of sets} (Corollary \ref{C:Stone}). Before these, we prove that the boundedness condition on the initial definition of {\wdl}s (Definition \ref{D:wdl}) is superfluous (Theorem \ref{T:wcl}, see also \cite{Kw09}).Comment: 15 page

Space complexity in on-line computation

AbstractA technique is developed for determining space complexity in on-line computation. It is shown that each of the following functions requires linear space: (i) the conversion of binary numbers into ternary numbers, (ii) the multiplication of integers and (iii) the translation of arithmetic expressions in infix notation into Polish notation

Some Examples of Minimal Groupoids on a Finite Set (Algebraic System, Logic, Language and Related Areas in Computer Science)

A minimal clone is an atom in the lattice of clones. The classification of minimal clones on a finite set still remains unsolved. A minimal groupoid is a minimal clone generated by a binary idempotent function. In this paper we report some examples of minimal groupoids generated by binary functions which resemble projections

On the isomorphism problem of concept algebras

Weakly dicomplemented lattices are bounded lattices equipped with two unary operations to encode a negation on concepts. They have been introduced to capture the equational theory of concept algebras (Wille 2000; Kwuida 2004). They generalize Boolean algebras. Concept algebras are concept lattices, thus complete lattices, with a weak negation and a weak opposition. A special case of the representation problem for weakly dicomplemented lattices, posed in Kwuida (2004), is whether complete weakly dicomplemented lattices are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem4). We also provide a new proof of a well known result due to M.H. Stone(Trans Am Math Soc 40:37-111, 1936), saying that each Boolean algebra is a field of sets (Corollary4). Before these, we prove that the boundedness condition on the initial definition of weakly dicomplemented lattices (Definition1) is superfluous (Theorem1, see also Kwuida (2009)

A Note on Idempotent Monomial Clones : Two is Strong; One is Weak (Developments of Language, Logic, Algebraic system and Computer Science)

Clones of polynomials are considered over Galois field GF(k). In particular, the class of clones generated by 2-variable idempotent polynomials is the target of our study. Our results include that the clone generated by x^{2}y^{k-2} is the largest among all such clones and the clone generated by xy^{k-1} is the smallest among all such clones. Hence, observing the exponent of one variable, two is strong and one is weak

A Study on Centralizing Monoids with Majority Operation Witnesses

A centralizing monoid M is a set of unary operations which commute with some set F of operations. Here, F is called a witness of M . On a 3-element set, a centralizing monoid is maximal if and only if it has a constant operation or a majority minimal operation as its witness. In this paper, we take one such majority operation, which corresponds to a maximal centralizing monoid, on a 3-element set and obtain its generalization, called mb , on a k-element set for any k >= 3. We explicitly describe the centralizing monoid M(mb ) with mb as its witness and then prove that it is not maximal if k > 3, contrary to the case for k = 3