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On the structure of linearly ordered pseudo-BCK-algebras. (English) Zbl 1182.06009
It is a nice, very interesting paper.
First, the authors recall the notions of (bounded) porim, integral residuated lattice, pseudo-BCK algebra, pseudo-BCK(pP) algebra, pseudo-hoop, pseudo-BL algebra, basic pseudo-hoop, Wajsberg pseudo-hoop, pseudo-MV algebra, cone algebra, pseudo-ŁBCK algebra and their connections, in order to establish the auxiliary result that for every (linearly ordered) cone algebra A there exists a bounded (linearly ordered) Wajsberg pseudo-hoop M such that A can be embedded into \((M, \rightarrow, \rightsquigarrow,1)\).
Then, the authors introduce the concept of cut of a linearly ordered pseudo-BCK algebra and the identities (H) and (J), that play a central role in the paper,
\[ \begin{aligned} (x \rightarrow y) \rightarrow (x \rightarrow z)&= (y \rightarrow x) \rightarrow (y \rightarrow z), \tag{H}\\ (((x \rightarrow y) \rightsquigarrow y) \rightarrow x) \rightsquigarrow x&= (((y \rightsquigarrow x) \rightarrow x) \rightsquigarrow y) \rightarrow y, \tag{J} \end{aligned} \] in order to characterize those linearly ordered pseudo-BCK algebras that arise as ordinal sums of linearly ordered cone algebras.
As applications, the authors study the \(\{\rightarrow, \rightsquigarrow, 1\}\)-subreducts of representable pseudo-hoops and pseudo-BL algebras, and they study the Bosbach states and the Riečan states on bounded linearly ordered pseudo-BCK algebras.
Reviewer’s remarks:
Pseudo-BCK algebras are more naturally connected to pseudo-Wajsberg algebras rather than to pseudo-MV algebras.
Identity (H) appears in the definition of L-algebras given in [W. Rump, “\(L\)-algebras, self-similarity, and \(l\)-groups”, J. Algebra 320, No. 6, 2328–2348 (2008; Zbl 1158.06009)].

MSC:
06F35 BCK-algebras, BCI-algebras (aspects of ordered structures)
03G12 Quantum logic
06D35 MV-algebras
Software:
Pseudo Hoops
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