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The present paper is a survey on a new kind of algebras, called EMV-algebras, generalizing both MV-algebras and generalized Boolean algebras. The survey is based on our papers [10, 11, 12, 13]. For these algebras a top element is not assumed a priori. Every EMV-algebra can be covered by a system of MV-algebras where each of them uses the restriction of \(\oplus ,\vee ,\wedge \). We prove that every such an EMV-algebra without top element can be embedded into an EMV-algebra with top element as its maximal ideal, and every EMV-algebra with top element is termwise equivalent to an MV-algebra. We show that the classes of EMV-algebras are intimately connected with subvarieties of MV-algebras. We establish a categorical equivalence of the category of EMV-algebras without top element with a special category of MV-algebras. We will study states and state-morphisms, their topological properties, a Krein–Mil’man-type representation, and an integral representation. We show also a kind of the Loomis–Sikorski theorem. Finally, we present some results on free EMV-algebras.

Some MV-algebras generated by families of fuzzy sets are considered. On the fuzzy MV-algebras a representation theorem is presented as well as a theorem of an invariant state in the case that the basic set is a locally compact topological group.

Starting from a classical theorem of Gurevich and Kokorin we survey recent diverging developments of the theories of lattice-ordered abelian groups and their counterparts equipped with a distinguished order unit. We will focus on decision and recognition problems. As an application of Elliott’s classification, we will touch on word problems of AF C*-algebras.

This paper gives new properties of WBR\(_0\)-algebras and commutative WBR\(_0\)-algebras. A comprehensive characterization for commutative WBR\(_0\)-algebras by various other logic algebras is obtained. Three characterizations with simpler types and fewer axioms of commutative WBR\(_0\)-algebras are also given.

In this paper we summarize results on perfect MV-algebras and we show that the class of first order A Lukasiewicz formulas valid in all interpretations over perfect MV-chains is axiomatizable.

Jan Łukasiewicz argued in his address which he delivered as a Rector of Warsaw University at the Inaguration of the academic year 1922/1923 [1] that “All sentences about future facts that are not yet decided belong to this [many-valued] category. Such sentences are neither true at the present moment ... nor are they false...”.

In this survey we give an account of the relationships between \(\ell \)-groups and some important algebraic structures like MV-algebras, BL-algebras, and their non-commutative versions given by pseudo MV-algebras and pseudo BL-algebras. In a similar way we show how partially ordered groups are connected with quantum structures like orthomodular lattices, effect algebras and pseudo effect algebras. For the latter classes, an important role is played by various types of the Riesz Decomposition Property.

The classical condition “c is a semantic consequence of a set P of premises” in infinite-valued propositional Łukasiewicz logic Ł\(_{\infty }\) is refined using enriched valuations that take into account the effect on the formula c of the stability of the truth-value of each formula p in P under small perturbations (or, measurement errors) of the models of P. The differential properties of the functions represented by c and by all p in P naturally lead to a new notion of semantic consequence that turns out to coincide with syntactic consequence.

In this chapter we present some specific properties of other non-commutative multiple-valued logic algebras: pseudo-MTL algebras, bounded Rl-monoids, pseudo-BL algebras and pseudo-MV algebras. As main results, we extend to the case of pseudo-MTL algebras some results regarding the prime filters proved for pseudo-BL algebras. The Glivenko property for multiple-valued logic algebras is defined and studied.

In this chapter we will present the notion of state for the case of pseudo-BCK algebras. One of the main results consists of proving that any Bosbach state on a good pseudo-BCK algebra is a Riecan state, but conversely it turns out not to be true. Some conditions are given for a Riecan state on a good pseudo-BCK algebra to be a Bosbach state. In contrast to the case of pseudo-BL algebras, we show that there exist linearly ordered pseudo-BCK algebras having no Bosbach states and that there exist pseudo-BCK algebras having normal deductive systems which are maximal, but having no Bosbach states. Some specific properties of states on FL w -algebras, pseudo-MTL algebras, bounded Rl-monoids and subinterval algebras of pseudo-hoops are proved. A special section is dedicated to the existence of states on the residuated structures, showing that every perfect FL w -algebra admits at least a Bosbach state and every perfect pseudo-BL algebra has a unique state-morphism. Finally, we introduce the notion of a local state on a perfect pseudo-MTL algebra and we prove that every local state can be extended to a Riecan state.

In this paper we shall deal with an extension of Łukasiewicz propositional logic obtained by considering scalar multiplication with real numbers, and we focus on the description of its Lindenbaum algebra, i.e., the algebra of truth functions. We show the correspondence between truth tables of such logic and multilayer perceptrons in which the activation function is the truncated identity.