Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 1, pp. 185-202 (2001)

On Some Influence of the Weak Subalgebra Lattice on the Subalgebra Lattice

Konrad Pioro

Institute of Mathematics, Warsaw University, ul. Banacha 2, PL-02-097 Warsaw, Poland; e-mail:

Abstract: In [P] we showed that for each locally finite unary (total) algebra of finite type, its weak subalgebra lattice uniquely determines its (strong) subalgebra lattice. Now we generalize this fact to algebras having also finitely many binary operations (for example, groupoids, semigroups, semilattices). More precisely, we generalize some ideas from [P] to prove: Let $\bf A$ be a locally finite (total) algebra with $m$ unary operations $k^ A_1,\ldots,k^ A_m$ and $n$ binary operations $f^ A_1,\ldots,f^ A_n$ and let $\bf A$ satisfy the following formula: for any $x,y$ and $1\le i\le n$, $x\neq y$ implies $f_i(x,y)\neq x$ and $f_i(x,y)\neq y$. Then for every partial algebra $\bf B$ with $m$ unary and $n$ binary operations, if the weak subalgebra lattices of $\bf A$ and $\bf B$ are isomorphic, then their (strong) subalgebra lattices are also isomorphic and moreover, $\bf B$ is total and locally finite and satisfies the same formula.

[P] Pioro, K.: On a strong property of the weak subalgebra lattice. Alg. Univ. 40 (1998), 477-495.

Keywords: hypergraph, strong and weak subalgebras, subalgebra lattices, partial algebra

Classification (MSC2000): 05C65, 05C99, 08A30; 05C90, 08A55

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