# zbMATH — the first resource for mathematics

On some unary algebras and their subalgebra lattices. (English) Zbl 1141.08004
For a partial unary algebra $$A$$ let $$S_s(A)$$, $$S_w(A)$$, $$G(A)$$ be the lattice of all strong subalgebras, the lattice of all weak subalgebras, and the digraph of $$A$$, respectively, and similarly for a poset $$(P,\leqq )$$. Further, for a lattice $$L$$, Ir$$(L)$$ be the set of all completely join-irreducible elements, $$D(L)=G($$Ir$$(L),\leqq )$$. The notion of a normality for digraphs, lattices and algebras is introduced. To each normal digraph $$G$$ there is associated a digraph $$TQ(G)$$. It is proved that if $$G$$ is a normal digraph and $$L$$ is a normal lattice, then $$S_s(G)\cong L\iff TQ(G)\cong D(L)$$.
The main result is as follows: Let $$A, B$$ be normal partial unary algebras and $$L$$ be a normal lattice. Then (a) $$S_s(A)\cong L\iff TQ(A)\cong D(L)$$, (b) $$S_s(A)\cong S_s(B)\iff TQ(A)\cong TQ(B)$$.
Applying the above result, necessary and sufficient conditions for lattices $$K, L$$ are found under which there is a normal partial unary algebra $$A$$ such that $$S_w(A)\cong K$$ and $$S_s(A)\cong L$$.

##### MSC:
 08A60 Unary algebras 08A30 Subalgebras, congruence relations 08A55 Partial algebras
Full Text:
##### References:
 [1] BARTOL W.: Weak subalgebra lattices. Comment. Math. Univ. Carolin. 31 (1990), 405-410. · Zbl 0711.08007 [2] BARTOL W.: Weak subalgebra lattices of monounary partial algebras. Comment. Math. Univ. Carolin. 31 (1990), 411-414. · Zbl 0711.08007 [3] BERGE C.: Graphs and Hypergraphs. North-Holland, Amsterdam, 1973. · Zbl 0254.05101 [4] BARTOL W.-ROSSELLÓ F.-RUDAK L.: Lectures on Algebras, Equations and Partiality. (F. Rosselló, Technical Report B-006, Dept. Ciencies Mat. Inf., Univ. Illes Balears, 1992. [5] BURMEISTER P.: A Model Theoretic Oriented Approach To Partial Algebras. Introduction to Theory and Application of Partial Algebras. Part I. Math. Res. 32, Akademie-Verlag, Berlin, 1986. · Zbl 0598.08004 [6] CRAWLEY P.-DILWORTH R. P.: Algebraic Theory of Lattices. Prentice Hall Inc, Englewood Cliffs, NJ, 1973. · Zbl 0494.06001 [7] JÓNSSON B.: Topics in Universal Algebra. Lecture Notes in Math. 250, Springer-Verlag, New York, 1972. · Zbl 0225.08001 [8] MCKENZIE R. N.-MCNULTY G. F.-TAYLOR W. F.: Algebras, Lattices, Varieties, Vol. I. Wadsworth & Brooks/Cole Math. Ser., Wadsworth Sc Brooks/Cole Advance Books &: Software, Montereu, California, 1987. · Zbl 0611.08001 [9] PIÓRO K.: On some non-obvious connections between graphs and partial unary algebras. Czechoslovak Math. J. 50(125) (2000), 295-320. · Zbl 1046.08002 [10] PIÓRO K.: On the subalgebra lattice of unary algebras. Acta Math. Hungar. 84 (1999), 27-45. · Zbl 0988.08004 [11] PIÓRO K.: On a strong property of the weak subalgebra lattice. Algebra Universalis 40 (1998), 477-495. · Zbl 0936.08002 [12] ORE O.: Theory of Graphs. Amer. Math. Soc. Colloq. Publ. 38, Amer. Math. Soc, Providence, RI, 1962. · Zbl 0105.35401 [13] ROBBINS H. E.: A theorem on graphs with application to a problem of traffic. Amer. Math. Monthly 46 (1939), 281-283. · Zbl 0021.35703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.