## Representation of integral quantales by tolerances.(English)Zbl 1471.06009

A quantale is a triple $$Q=(Q,\bigvee,\otimes)$$, where $$(Q,\bigvee)$$ is a $$\bigvee$$-semilattice (partially ordered set having arbitrary joins), and $$(Q,\otimes)$$ is a semigroup such that the operation $$\otimes$$ distributes over arbitrary joins from both sides, namely, $$a\otimes(\bigvee S)=\bigvee_{s\in S}(a\otimes s)$$ and $$(\bigvee S)\otimes a=\bigvee_{s\in S}(s\otimes a)$$ for every element $$a\in Q$$ and every subset $$S\subseteq Q$$ [K. I. Rosenthal, Quantales and their applications. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. (1990; Zbl 0703.06007)]. A quantale $$Q$$ is said to be unital provided that its underlying semigroup $$(Q,\otimes)$$ is a monoid, namely, has a unit element $$e$$. A unital quantale $$Q$$ is integral provided that its unit is the top element $$\top_Q$$ of the $$\bigvee$$-semilattice $$(Q,\bigvee)$$. For example, for every unital quantale $$Q$$, the set $$A=\{a\in Q\,|\,a\leqslant e\}$$ is a subquantale of $$Q$$, which is an integral quantale (the integral part of $$Q$$).
There exist different representation theorems for quantales. For example, [S. Valentini, Math. Log. Q. 40, No. 2, 182–190 (1994; Zbl 0816.06018)] showed that every quantale $$Q$$ is isomorphic to the quantale of the so-called ordered relations on $$Q$$. Taking on this result, the present paper provides a “more transparent” (as claimed by the authors) description of ordered relations and also shows how the reflexive ordered relations are connected to tolerances (reflexive and symmetric binary relations, compatible with the respective algebraic structure on the underlying set). Moreover, the authors prove that every integral quantale $$Q$$ has a natural embedding into the quantale of complete tolerances on the underlying lattice of $$Q$$.
The paper additionally shows that the underlying lattice of any finite integral quantale $$Q$$ is dually pseudocomplemented and distributive in $$\top_Q$$, where a lattice $$L$$ is said to be pseudocomplemented provided that for every $$a\in L$$, there exists an element $$a^{*}\in L$$ (the pseudocomplement of $$a$$) such that for every $$b\in L$$, it follows that $$b\wedge a=\bot_L$$ if and only if $$b\leqslant a^{*}$$.
Lastly, the authors prove that certain relations on different algebraic structures naturally form a quantale, e.g., the set of all compatible reflexive binary relations on every finite majority algebra makes an integral quantale (we recall that a ternary term $$m$$ of an algebra $$A$$ is a majority term provided that the following identities hold on $$A$$: $$m(x,x,y)=m(x,y,x)=m(y,x,x)=x$$; for example, every algebra with a lattice reduct admits such a majority term; an algebra $$A$$ admitting a majority term is called a majority algebra).
The paper is extremely well written, provides the most essential parts of its required preliminaries, and will be of interest to all those researchers who study the theory of quantales and especially their representation theorems.

### MSC:

 06F07 Quantales 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06B15 Representation theory of lattices 06B23 Complete lattices, completions 06D22 Frames, locales

### Citations:

Zbl 0703.06007; Zbl 0816.06018
Full Text:

### References:

  Bandelt, H-J, Local polynomial functions on lattices, Houst. J. Math., 7, 317-325, (1981) · Zbl 0479.06007  Bartl, E; Krupka, M, Residuated lattices of block relations: size reduction of concept lattices, Int. J. Gen. Syst., 45, 773-789, (2016) · Zbl 1404.68154  Blyth, TS, Residuated mappings, Order, 1, 187-204, (1984) · Zbl 0553.06001  Blyth, T.S., Janowitz, M.F.: Residuation Theory. Pergamon Press, Oxford (1972) · Zbl 0301.06001  Chajda, I; Radeleczki, S, $$0$$-conditions and tolerance schemes, Acta Math. Univ. Comenian. N.S., 72, 177-184, (2003) · Zbl 1087.08002  Czédli, G; Horváth, EK; Radeleczki, S, On tolerance lattices of algebras in congruence modular varieties, Acta Math. Hungar., 100, 9-17, (2003) · Zbl 1049.08007  Dilworth, RP; Ward, M, Residuated lattices, Trans. Am. Math. Soc., 45, 335-354, (1939) · Zbl 0021.10801  Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, vol. 151. Elsevier, Amsterdam (2007) · Zbl 1171.03001  Golan, J.: Semirings and Their Applications. Kluwer, Dordrecht (1999) · Zbl 0947.16034  Grätzer, G; Schmidt, ET, On the lattice of all join-endomorphisms of a lattice, Proc. Am. Math. Soc., 9, 722-726, (1958) · Zbl 0087.26104  Hobby, D., McKenzie, R.: The Structure of Finite Algebras. Contemporary Mathematics, vol. 76. American Mathematical Society, Providence (1988) · Zbl 0721.08001  Janowitz, MF, Decreasing Baer semigroups, Glasgow Math. J., 10, 46-51, (1969) · Zbl 0175.01703  Janowitz, MF, Tolerances and congruences on lattices, Czech. Math. J., 36, 108-115, (1986) · Zbl 0598.06004  Kaarli, K, Subalgebras of the squares of weakly diagonal majority algebras, Studia Sci. Math. Hungar., 49, 509-524, (2012) · Zbl 1289.08009  Kaarli, K., Pixley, A.: Polynomial Completenes in Algebraic Systems. CRC Press, Boca Raton (2000)  Kaarli, K; Kuchmei, V; Schmidt, SE, Sublattices of the direct product, Algebra Universalis, 59, 85-95, (2008) · Zbl 1170.06003  Mulvey, CJ; Hazewinkel, M (ed.), Quantales, (2001), Berlin  Rosenthal, K.I.: Quantales and Their Applications. Pitman Research Notes in Mathematics Series, vol. 234. Longman Scientific and Technical, Harlow (1990) · Zbl 0703.06007  Valentini, S, Representation theorems for quantales, Math. Logic Q., 40, 182-190, (1994) · Zbl 0816.06018  Varlet, JC, A generalization of the notion of pseudo-complementedness, Bull. Soc. R. Sci. Liêge, 37, 149-158, (1968) · Zbl 0162.03501  Wille, R, Über endliche ordnungsaffinvollständige verbande, Math. Z., 155, 103-107, (1977) · Zbl 0357.06007
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